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Appell sequences, continuous wavelet transforms and series expansions
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Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha
Appell sequences, continuous wavelet transforms and series expansions Say Song Goh a , Tim N.T. Goodman b , S.L. Lee a,∗ a
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, The University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
a r t i c l e
i n f o
Article history: Received 19 March 2013 Received in revised form 23 August 2015 Accepted 15 January 2016 Available online xxxx Communicated by Ding-Xuan Zhou Keywords: Appell sequences Scale-space Wavelet transforms Gaussian Hermite polynomials B-splines
a b s t r a c t A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives respectively, the associated Bernoulli polynomials converge to the Hermite polynomials, and the corresponding series expansion is an expansion in terms of the Hermite polynomials, the Gaussian scale-space and the continuous wavelet transforms with the derivatives of the Gaussian function as mother wavelets. A similar expansion is also derived in terms of continuous wavelet transforms in which the mother wavelets are the spline framelets that approximate the derivatives of the standardized B-splines. © 2016 Published by Elsevier Inc.
1. Introduction The object of this paper is to present some new results involving the classical Bernoulli and Hermite polynomials, and to connect scale-space and wavelet transforms with classical series expansions of realvalued functions in terms of these Appell polynomials. A brief review and a general approach to scale-space and continuous wavelet transforms in the context of singular integral operators can be found in [2]. The Bernoulli polynomials are associated with the uniform B-splines in the same way as the Hermite polynomials are associated with the Gaussian function. The main result is Theorem 1.1 below, but in order to give a systematic presentation, we shall begin with the general setting and consider a compactly supported probability measure or probability density function μ, of which the standardized uniform B-splines are examples. Then for any integer m ≥ 0, * Corresponding author. E-mail addresses: [email protected] (S.S. Goh), [email protected] (T.N.T. Goodman), [email protected] (S.L. Lee). http://dx.doi.org/10.1016/j.acha.2016.01.005 1063-5203/© 2016 Published by Elsevier Inc.
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μ(m) , e(·)z = (−1)m μ, z m e(·)z = (−1)m z m μ (iz) and so m (m)
(−1) μ
e(·)z , μ (iz)
= zm
(1.1)
in a neighborhood of 0, where the derivatives μ(m) are in the sense of distribution, ·, · denotes the action ∞ of distributions on test functions, and μ is the Fourier transform defined by μ (u) := −∞ e−iut dμ(t). Since μ is compactly supported, μ is analytic, and so we can define a sequence of Appell polynomials Pμ,m by the generating function ∞ exz Pμ,m (x) m = z , μ (iz) m=0 m!
which, together with (1.1), gives the biorthogonal relation Pμ,m = δm,n . (−1)n μ(n) , m!
(1.2)
∞ The biorthogonal system {(−1)m μ(m) }∞ m=0 , {Pμ,m /m!}m=0 provides a formal biorthogonal expansion for ∞ f ∈ C of the form
f (x) =
∞
(−1)m μ(m) , f
m=0
∞ Pμ,m (x) Pμ,m (x) = . μ, f (m) m! m! m=0
(1.3)
Such expansions include a large class of classical series expansions, such as Taylor’s series, the Euler– Maclaurin expansion and the Lidstone series [8]. For the case where μ is the Gaussian density function 2 G(t) := √12π e−t /2 , which is not compactly supported but the above procedure is applicable, the corresponding Appell sequence comprises the Hermite polynomials, the biorthogonal relation (1.2) is the orthonormal relation with respect to the weighted inner product with Gaussian weight and the expansion (1.3) is the classical Hermite polynomial expansion. This paper focuses on the case where the probability density functions are the standardized B-splines and derives an expansion of the form (1.3) for functions in a Sobolev space. To this end, we define for ν > 1, the Sobolev space H ν (R) that comprises all tempered distributions f for which f 2H ν
1 := 2π
∞
|f(u)|2 (1 + |u|2 )ν du < ∞.
−∞
For the forward uniform B-splines Mn of order n, the Fourier transforms are n (u) = M
1 − e−iu iu
n ,
u ∈ R,
and so n (iz) = M
ez − 1 z
n ,
z ∈ C.
The corresponding Appell sequences of polynomials, which we denote by Bn,m , are the Bernoulli polynomials of order n generated by
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3
|z| < 2π.
n As a probability distribution, the mean and variance of Mn are μn = n2 and σn2 = 12 respectively. Let n (x) := σn Mn (σn x + μn ) be the standardized B-splines with mean 0 and variance 1. Then M
n (u) = eiun/(2σn ) M
1 − e−iu/σn iu/σn
n u ∈ R.
,
Further the standardized Bernoulli polynomials n,m (x) := 1 Bn,m (σn x + μn ), B σnm are generated by the generating function ∞ exz Bn,m (x) m z , = m! n (iz) m=0 M
|z| < 2πσn ,
(1.4)
where for every x ∈ R, the convergence is uniform on |z| ≤ ρ, for any ρ < 2πσn . The main result is Theorem 1.1. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then for x, t ∈ R, s > 0, f (x) = SM n f (s, t) +
n−1
n,m (s(t − x)) (−1)m B WM (m) f (s, t) + Rn f (x, s, t), n m! m=1
(1.5)
where Rn f (x, s, t) → 0 as n → ∞ locally uniformly for x ∈ R and s > 0 and uniformly for t ∈ R, and ∞ SM n f (s, t) :=
n (s(t − x))f (x)dx, sM
t ∈ R, s > 0,
−∞
is the B-spline scale-space transform of f , ∞ WM (m) f (s, t) := n
(m) (s(t − x))f (x)dx, sM n
t ∈ R, s > 0,
−∞
are continuous wavelet transforms of f with the derivatives of the standardized B-splines as mother wavelets. The representation of f (x) in (1.5) holds for any t ∈ R and s > 0. In particular, if t = 0 and s = 1, it becomes f (x) =
n−1
(m) , f (−1)m M n
m=0
n,m (x) B + Rn f (x), m!
where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R, which is the expansion of f , with remainder, in n,m /m!}. On the other hand, if t = x, then (1.5) gives n(m) }, {B terms of the biorthogonal system {(−1)m M f (x) = SM n f (s, x) +
n−1
n,m (0) (−1)m B WM (m) f (s, x) + Rn f (s, x), n m! m=1
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which is a decomposition of f into its B-spline scale-space and wavelet transforms. Since spline functions were introduced by Schoenberg [4], such an expansion involving spline functions and Bernoulli polynomials will be referred as the Bernoulli–Schoenberg series. To prove Theorem 1.1, we first develop the corresponding expansion where the Gaussian function G(t) = 2 √1 e−t /2 defines the Gaussian scale-space 2π ∞ sG(s(t − x))f (x)dx,
SG f (s, t) :=
t ∈ R, s > 0,
−∞
and its derivatives G(m) , m = 1, 2, . . . , are mother wavelets that define the continuous wavelet transforms ∞ sG(m) (s(t − x))f (x)dx,
WG(m) f (s, t) :=
t ∈ R, s > 0.
−∞
The Gaussian function also defines the Appell sequence of Hermite polynomials Hm , m = 0, 1, . . . , by the generating function ∞ exz Hm (x) m z , = G(iz) m=0 m!
(1.6)
2 where G(u) = e−u /2 is the Fourier transform of G, and for every x ∈ R, the series converges locally uniformly on C. Also (−1)m G(m) (t) = Hm (t)G(t) and Hm /m! are biorthogonal sequences, which is the well-known √ fact that the normalized Hermite polynomials Hm / m!, m = 0, 1, . . . , form an orthonormal basis for the ∞ Hilbert space L2G (R) with inner product f, gG := −∞ f (t)g(t)G(t)dt, and so for every f ∈ L2G (R),
∞
f (x) =
f, Hm G
m=0
Hm (x) , m!
where the convergence is in L2G (R). We shall show that if f ∈ H ν (R) for some ν > 1 and is continuous, then f (x) = SG f (s, t) +
∞ Hm (s(x − t)) WG(m) f (s, t), m! m=1
(1.7)
where the series converges uniformly in t ∈ R and locally uniformly in x ∈ R and s > 0. Note that if t = 0 and s = 1, the resulting series is the Hermite expansion in Sobolev space. On the other hand, since
Hm (0) =
0, (−1)j (2j)! , 2j j!
m = 2j + 1, m = 2j,
j = 0, 1, . . . ,
if t = x, the expansion becomes
f (x) = SG f (s, x) +
∞ (−1)j j=1
2j j!
WG(2j) f (s, x),
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which is a decomposition of f into its Gaussian scale-space and wavelet transforms. Therefore, (1.7) provides an expansion that unifies the classical Hermite polynomial expansion on one hand and Gaussian scale-space and wavelet decomposition on the other. Section 2 gives a proof of (1.7). The proof of Theorem 1.1 is based on equation (1.7) and requires precise estimates of the rate of convergence of the Fourier transforms of the derivatives of the standardized B-splines to the Fourier transforms of the derivatives of the Gaussian function, as well as on the rate of convergence of the standardized Bernoulli polynomials to the Hermite polynomials as the order of the B-spline tends to infinity. Section 3 studies the rate of convergence of the derivatives of the standardized B-splines, where we also show that for any nonnegative integer m and 0 < < 1, (m) (m) M − G = O n p
1
n1−/p
as n → ∞,
1 ≤ p ≤ ∞,
(m) a result which is of interest in itself. In particular, M − G(m) = O n1 . The convergence of the n ∞ derivatives of B-splines has been established in [1] and pointwise convergence of order O(1/n) is given in [7], where they also deduce Lp -convergence for p ∈ [2, ∞), but no convergence rate in uniform or Lp -norm is given. The rate of convergence of the standardized Bernoulli polynomials to the Hermite polynomials is given in Section 4 and the proof of Theorem 1.1 in Section 5. In Section 6 we show that when suitably standardized, the spline framelets of Ron and Shen [3] approximate the derivatives of the standardized B-splines and converge to the derivatives of the Gaussian function with the same rate as that of the derivatives of the B-splines. Bernoulli–Schoenberg series expansion formulas of the form (1.5) are then derived with the standardized spline framelets as mother wavelets in place of the derivatives of the B-splines in the wavelet transforms. 2. Hermite polynomial expansion, Gaussian scale-space and wavelet transforms The Gaussian function G(t) := √12π e−t /2 defines the Gaussian scale-space, and its derivatives G(m) , m = 1, 2, . . . , are mother wavelets that define continuous wavelet transforms. It also generates the Appell sequence of Hermite polynomials, Hm , m = 0, 1, . . . , by the generating function (1.6). Let L2G (R) be the ∞ Hilbert space with inner product f, gG := −∞ f gG. It is well known that G(m) = (−1)m Hm G, m = 0, 1, . . . , and 2
Hj , Hm /m!G = δjm ,
j, m = 0, 1, . . . .
Any f ∈ L2G (R) can be represented by
f=
∞
f, Hm G Hm /m!
(2.1)
m=0
in L2G (R). Letting f, g denote the usual L2 inner product, for any f ∈ L2 (R) ⊂ L2G (R), (2.1) implies that
f=
∞
(−1)m G(m) , f Hm /m!
m=0
in L2G (R). For t, x ∈ R and s > 0, let fs,t (x) := f (t − x/s). Then for m = 0, 1, . . . ,
(2.2)
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∞ (−1) G m
(m)
, fs,t = (−1)
G(m) (x)f (t − x/s)dx
m −∞
∞ m
= (−1)
sG(m) (s(t − y))f (y)dy,
−∞
and so G, fs,t = SG f (s, t)
(2.3)
(−1)m G(m) , fs,t = (−1)m WG(m) f (s, t).
(2.4)
and for m = 1, 2, . . . ,
Therefore, we have by (2.2),
fs,t = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm /m!
m=1
in L2G (R) for any t ∈ R and s > 0. To obtain uniform convergence we require f to belong to some Sobolev space. Here we are concerned with H ν (R) for some ν > 1, and in this case, H ν (R) ⊂ L2 (R). We shall prove Theorem 2.1. If f ∈ H ν (R) for some ν > 1 and is continuous, then for any t, x ∈ R and s > 0, f (t − x/s) = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm (x)/m!,
(2.5)
m=1
where the convergence is uniform over t ∈ R and locally uniform over s > 0 and x ∈ R. Remark 1. Since Hm (t) = (−1)m Hm (−t), a change of variable in (2.5) gives the following equivalent representation
f (x) = SG f (s, t) +
∞ Hm (s(x − t)) WG(m) f (s, t). m! m=1
First we show that under the conditions of Theorem 2.1, convergence is locally uniform in (2.1). The method and estimates in the proof of Theorem 2.2 below will also be used elsewhere in the paper. Theorem 2.2. If f ∈ H ν (R) for some ν > 1 and is continuous, then
f (x) =
∞
(−1)m G(m) , f Hm (x)/m!,
m=0
where the convergence is locally uniform over x ∈ R.
(2.6)
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Proof. For n = 0, 1, . . . , ∞
2
n
f (x)G(x) −
(−1) G m
(m)
m=0
−∞
∞
n
f (x) −
≤
(−1) G m
m=0
−∞
(m)
Hm (x) G(x) , f m!
Hm (x) , f m!
dx
2 G(x)dx
2 n m (m) = f − (−1) G , f Hm /m! . m=0
G
Now Hm G ∈ L2 (R), and since H ν (R) ⊂ L2 (R), f G also belongs to L2 (R), and by (2.2), this gives f (x)G(x) =
∞
(−1)m G(m) , f Hm (x)G(x)/m!
(2.7)
m=0
in L2 (R). We want to show that the series on the right of (2.7) converges uniformly. For m = 0, 1, . . . , x ∈ R, m
(−1) Hm (x)G(x) = G
(m)
1 (x) = 2π
∞
(iu)m e−u
2
/2 iux
e
du,
−∞
and so 1 |Hm (x)G(x)| ≤ π
∞
um e−u
2
/2
du =
2
m−1 2
π
Γ ((m + 1)/2) .
(2.8)
0
By the Stirling formula, √
Γ ((m + 1)/2) ∼
2π m m/2 , 2m/2 e
m! = Γ(m + 1) ∼
m m √ 2πm , e
and so for all sufficiently large m, 1 e m/2 |Hm (x)G(x)| ≤ K√ . m! m m
(2.9)
Also for m = 0, 1, . . . , ∞ (−1) G m
(m)
, f = (−1)
m
(m)
f (t)G
−∞
(−1)m (t)dt = 2π
∞
2 f(u)(−iu)m e−u /2 du
−∞
and so for some constant C > 0,
|(−1) G m
(m)
1 , f | ≤ 2π
∞ −∞
|f(u)||u|ν |u|m−ν e−u
2
/2
du
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⎛ ≤ Cf H ν ⎝
∞
⎞1/2 |u|2m−2ν e−u du⎠ 2
−∞
⎛
= Cf H ν ⎝2
∞
⎞1/2 u2m−2ν e−u du⎠ 2
0 1/2
= Cf H ν Γ (m − ν + 1/2)
.
(2.10)
Again by the Stirling formula, Γ (m − ν + 1/2) ∼
2π(m − ν − 1/2)
m − ν − 1/2 e
m−ν−1/2 ∼
√
2π
1 m m , mν e
and so (2.10) gives |(−1)m G(m) , f | ≤ C
f H ν m m/2 , e mν/2
(2.11)
where the constant C is a generic constant independent of m. It follows from (2.9) and (2.11) that f H ν (−1)m G(m) , f Hm (x)G(x)/m! ≤ C (1+ν)/2 , m
(2.12)
for all x ∈ R. Since ν > 1, we conclude that the series in (2.7) converges uniformly on R. It follows that (2.6) holds locally uniformly on R. 2 Remark 2. Although Theorem 2.1 involves classical Hermite polynomial expansion, we are unable to find similar results in the literature. More information on local uniform convergence of Hermite polynomial expansions can be found in the classic monograph of Szegö ([6], Theorem 9.1.6). Proof of Theorem 2.1. For any t ∈ R and s > 0, let fs,t (x) := f (t − x/s), x ∈ R. Then fs,t ∈ H ν (R) and its Fourier transform is given by fs,t (u) = sf(−us)e−iust , u ∈ R. By Theorem 2.2, for t ∈ R and s > 0, fs,t (x) =
∞
(−1)m G(m) , fs,t Hm (x)/m!,
(2.13)
m=0
where the convergence is locally uniform over x ∈ R. A calculation similar to (2.10) shows that for some constant C > 0, (−1)m G(m) , fs,t ≤ C
1 sν−1/2
f H ν Γ(m − ν + 1/2)1/2 ,
t ∈ R, s > 0.
Proceeding as in the proof of Theorem 2.2, we see that the convergence of the series in (2.13) is uniform over t ∈ R and locally uniform over s > 0. Combining with (2.3) and (2.4) gives (2.5). 2 Corollary 2.3. If for some ν > 1, f ∈ H ν (R) and is continuous, then f (t) = SG f (s, t) +
∞ (−1)j j=1
2j j!
WG(2j) f (s, t),
where the convergence is uniform over t ∈ R and locally uniform over s > 0.
(2.14)
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Proof. Since f is continuous, we may set x = 0 in (2.5) to give f (t) = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm (0)/m!,
(2.15)
m=1
where the convergence is uniform over t ∈ R and locally uniform over s > 0. Now 1 G(m) (0) = (−1)m Hm (0)G(0) = √ (−1)m Hm (0), 2π and since G(m) (0) = 0 if m is odd and G(2j) (0) =
(−1)j (2j)! √1 , 2j j! 2π
(2.14) follows from (2.15). 2
3. Convergence of derivatives of B-splines We shall first prove some lemmas, which will be used to derive the rate of convergence of the derivatives of the standardized B-splines in the frequency domain as well as in the proof of the main theorem. Lemma 3.1. For |u| ≤ nα , 0 < α < 1/2, n (u) ≤ e−u2 /2 0≤M
(3.1)
and for all sufficiently large n, 4 −u2 /2 n (u) − e−u2 /2 ≤ K u e M , n where K is an absolute constant. Proof. By Taylor’s theorem, for |u| ≤ n (u) = M
n/3 π,
n
n sin u 3/n 3u4 u2 + ≤ 1− . 2n 40n2 u 3/n
Now for 0 ≤ s ≤ 6/5, e−s ≥ 1 − s + s2 /2 − s3 /6 ≥ 1 − s + 3s2 /10, and so for u2 ≤ 12n/5, 1−
2 3u4 u2 + ≤ e−u /2n , 2n 40n2
which gives (3.1). 6 u2 u4 u n (u) = e− 2 − 20n +O n2 , and so Also M 4 6 6 u4 u n (u) − e−u2 /2 = e−u2 /2 1 − e− 20n +O n2 ≤ e−u2 /2 u + O u M 20n n2 2 2 2 e−u /2 |u|4 1 |u|4 e−u /2 u = 20 + O n ≤ K n n for |u| ≤
√
n for all sufficiently large n.
2
(3.2)
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Lemma 3.2. Take integers m > 0, n > m and 0 < α < 1/2. Then for all sufficiently large n, m n (u) ≤ nmα e− 12 n2α for u ≥ nα . u M Proof. Let Kn,m
Kn,m
(3.3)
m := supu≥nα u M n (u). Then
n n m/2 n m/2 | sin x|n m sin u 3/n = sup u = Ann,m , = √sup n(1−m/n) x≥ 3nα−1/2 3 3 x u 3/n u≥nα
where An,m :=
sup
√ x≥ 3nα−1/2
| sin x| . x1−m/n
Observe that | sin x| sin x = sup , 1−m/n 1−m/n x≥π/2 x π/2≤x≤π x sup
and so for
√ α−1/2 3n ≤ π/2, An,m = √
Consider the function g(x) :=
sin x , x1−m/n
sin x
sup 3nα−1/2 ≤x≤π
x1−m/n
.
0 ≤ x ≤ π, where g(0) = g(π) = 0 and
g (x) = {x cos x − (1 − m/n) sin x}/x2−m/n = cos x{x − (1 − m/n) tan x}/x2−m/n . Then g (x) = 0 only at the unique value x = ζ, 0 < ζ < π/2, where tan ζ = ζ/(1 − m/n). We want to show √ that ζ < 3nα−1/2 . Consider m tan x − x 1 m m − 1− . x− 1− tan x = 3 2 x n nx n x3 At x =
√
3nα−1/2 , as n → ∞, x → 0 and nx2 = 3n2α → ∞. Thus
1 1 tan x − x m =− . x − 1 − tan x = − lim 3 3 n→∞ x x→0 n x 3 √ √ Therefore for all large enough n, g ( 3nα−1/2 ) < 0. Since g (π) < 0, ζ < 3nα−1/2 . Hence g is decreasing √ on [ 3nα−1/2 , π]. Thus lim
An,m = √
sin x
sup 3nα−1/2 ≤x≤π
x1−m/n
√ √ α−1/2 sin( 3nα−1/2 ) ) m/2n m(α−1/2)/n sin( 3n √ = √ =3 n , α−1/2 1−m/n α−1/2 ( 3n ) 3n
and so Kn,m =
n m/2 3
Ann,m
mα
=n
n sin(nα 3/n) . nα 3/n
(3.4)
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By (3.1) of Lemma 3.1 with u = nα ,
n sin(nα 3/n) 1 2α ≤ e− 2 n . nα 3/n
It follows from (3.4) that Kn,m ≤ nmα e− 2 n 1
2α
for all large enough n. 2
Theorem 3.3. For any integer m ≥ 0,
m 1 m −u2 /2 =O as n → ∞. sup u M n (u) − u e n u∈R
(3.5)
Proof. For any nonnegative integer m, (3.2) gives m+4 −u2 /2 m e n (u) − um e−u2 /2 ≤ K |u| u M , n
|u| < nα , 0 < α < 1/2.
The right hand side of the inequality attains its maximum at |u| = It follows that for all |u| < nα ,
√
m + 4 < nα for all sufficiently large n.
(m+4)/2
m 1 n (u) − um e−u2 /2 ≤ K m + 4 u M . =O n e n Since |u|m e−u
2
/2
(3.6)
is decreasing for all large enough |u|, we see that for all large enough n
1 , n
|u| ≥ nα .
mα m n (u) ≤ n2α = O 1 , u M en /2 n
|u| ≥ nα .
|um e−u
2
/2
|≤
nmα =O en2α /2
(3.7)
Also from (3.3), for all sufficiently large n,
Therefore,
m n (u) − um e−u2 /2 ≤ um M n (u) + |um e−u2 /2 | = O 1 , u M n
|u| ≥ nα .
By (3.6) and (3.8), we obtain (3.5). 2 Theorem 3.4. For any integer m ≥ 0, (m) Mn − G(m)
∞
1 as n → ∞. =O n
Proof. By taking inverse Fourier transforms, for n ≥ 1 and x ∈ R, (m) (x) − G(m) (x) = 1 M n 2π and so
∞ e −∞
iux
(iu)
m
−u2 /2 du M n (u) − e
(3.8)
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12
∞ 1 (m) (m) n (u) − e−u2 /2 du. um M Mn (x) − G (x) ≤ π
(3.9)
0
By (3.2), for 0 < α < 1/2, n
α n 2 2 K −u /2 n (u) − e du ≤ u M um+4 e−u /2 du n
α
m
0
0
K ≤ n
∞ u
m+4 −u2 /2
e
1 . du = O n
(3.10)
0
Take any integer > 1. Then by (3.3) of Lemma 3.2 with m replaced by m + , for sufficiently large n, (m+)α m n (u) ≤ n1 u M e 2 n2α u ,
u ≥ nα , 0 < α < 1/2.
Also, as in the proof of (3.7), |um e−u
2
/2
|≤
n(m+)α , 1 2α e 2 n u
u ≥ nα ,
and so ∞
∞ ∞ (m+)α m m −u2 /2 2 n −u /2 n (u) − e n (u) + u e u M du ≤ u M du du ≤ 2 1 2α e 2 n u m
nα
nα
nα
2n(m+)α 2n(m+1)α = = 1 2α 1 2α = O ( − 1)e 2 n n(−1)α ( − 1)e 2 n
1 . n
(3.11)
The result follows from (3.9), (3.10) and (3.11). 2 Theorem 3.5. Take any integer m ≥ 0. Then for any 0 < < 1 and 1 ≤ p < ∞, (m) (m) Mn − G = O
1 n1−/p
p
as n → ∞.
(3.12)
Proof. By Theorem 3.4, we have (m) Mn − G(m)
∞
=O
1 . n
(3.13)
Then for any positive < 1 and for all n ≥ 1, n (m) (m) − G(m) Mn (x) − G(m) (x) dx ≤ 2n M n
−n
∞
≤
C , n1−
(3.14)
where C > 0 is a generic constant. Thus for all n ≥ 1, n n (m) (x)dx − G(m) (x)dx ≤ C . M n n1− 0
0
(3.15)
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Now for m even,
∞ 0
∞
n(m) (x)dx = M
(m) (x)dx − M n
0
∞ 0
G(m) (x)dx, and for m odd, by Theorem 3.4,
∞ G
13
(m)
(m−1) (0) + G(m−1) (0) = O (x)dx = −M n
1 , n
0
and so ∞ ∞ (m) (x)dx − G(m) (x)dx ≤ C . M n n 0
(3.16)
0
It follows from (3.15) and (3.16) that ∞
∞ 1 (m) (x)dx − G(m) (x)dx = O M . n n1− n
(3.17)
n
n(m) has exactly m simple Now G(m) has exactly m simple zeros, a1 < a2 < · · · < am , and similarly M n n n n zeros, a1 < a2 < · · · < am . Then from (3.13) limn→∞ aj = aj , j = 1, 2, . . . , m. So for all large enough n, n(m) have no zero on [n , ∞), and therefore G(m) and M ∞
∞ 1 (m) G (x) dx = G(m) (x)dx = O n
n
(3.18)
n
and ∞ ∞
∞ 1 (m) (m) (m) , Mn (x) dx = Mn (x)dx = G (x)dx + O n1−
n
n
(3.19)
n
by (3.17). Then (3.14), (3.18) and (3.19) give
1 (m) . Mn − G(m) = O n1− 1 Now, using the inequality 1/p
1−1/p f p ≤ f ∞ f 1
for any f ∈ L1 (R) ∩ L∞ (R) and 1 < p < ∞, we have (3.12). 2 4. Convergence of Bernoulli polynomials to Hermite polynomials n,m to the Hermite polyTo obtain the rate of convergence of the standardized Bernoulli polynomials B nomials Hm as n → ∞, we note from (1.4) and (1.6) that ∞ exz exz Bn,m (x) − Hm (x) m z , − z2 /2 = m! n (iz) e m=0 M
and so by Cauchy’s integral formula,
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⎛ ⎞ 2 1 Bn,m (x) − Hm (x) 1 dz = exz ⎝ − e−z /2 ⎠ m+1 , m! 2πi z n (iz) M C where C is a circle with center at the origin and radius r < 2πσn = π
(4.1)
n/3.
Lemma 4.1. For z in a compact subset of C,
1 2 1 −z /2 as n → ∞. −e =O n M n (iz) Proof. Take n large enough so that the open disk with center at the origin and radius π given compact set. Then for z in the compact set, a direct computation gives ⎛ Log ⎝
⎞ 1 n (iz) M
2 4 ⎠+ z = z +O 2 20n
z6 n2
(4.2) n/3 contains the
,
and so ⎛ ⎞ 2 1 z Log ⎝ ⎠ + ≤ A. 2 n n (iz) M Therefore, ⎛ ⎞
2 n (iz) 1 Log 1/ M 2 2 1 −z /2 −z /2 Log ⎝ ⎠ + z ≤ KA , e = ≤ K − e − e 2 n M n (iz) n (iz) M where A, K are constants independent of n. 2 Proposition 4.2. For positive integers m < n and x ∈ R, 2 B e m/2 n,m (x) − Hm (x) Kex , ≤ m! n 2m
(4.3)
where K is a generic constant that is independent of m, n. Proof. By (4.1) and (4.2) followed by writing z = ζ + iη, ζ, η ∈ R, ⎛ ⎞ B 2 2 2 1 (x) − H (x) 1 dz n,m −x m xz−x ⎝ −z /2 ⎠ = e −e e m+1 m! z 2πi M n (iz) C xz−x2 xζ−x2 K K e e ≤ |dz| |dz| = 2πn z m+1 2πn rm+1 C
K ≤ 2πnrm+1
C
e−(x−ζ/2) eζ 2
2
/4
|dz|
C
≤
K 2πnrm+1
2
eζ C
2
/4
|dz| ≤
Ker /4 , nrm
(4.4)
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15
r 2 /4 for all positive r < π n/3, where r is the radius of the circle C. The minimum of e rm is attained at √ √ r = 2m, which is less than π n/3, since m < n. Therefore, (4.4) holds in particular for r = 2m, i.e. B K e m/2 n,m (x) − Hm (x) −x2 ≤ , e m! n 2m and so the result follows. 2 5. Proof of Bernoulli–Schoenberg series formula We first prove the following Theorem 5.1. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then
f (x) =
n−1
(m) , f (−1)m M n
m=0
n,m (x) B + Rn f (x), m!
(5.1)
where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R. Proof. We want to show that Rn f (x) := f (x) − uniformly for x ∈ R. By (2.6) of Theorem 2.2,
Rn f (x) =
∞
n−1
m (m) m=0 (−1) Mn , f Bn,m (x)/m!
(−1)m G(m) , f Hm (x)/m! −
m=0
n−1
→ 0 as n → ∞ locally
(m) , f B n,m (x)/m! (−1)m M n
m=0
= An f (x) + Bn f (x) +
∞
(−1)m G(m) , f Hm (x)/m!,
(5.2)
m=n
where
An f (x) :=
n−1
n(m) , f Hm (x)/m!, (−1)m G(m) , f − (−1)m M
(5.3)
m=0
Bn f (x) :=
n−1
n(m) , f Hm (x) − B n,m (x) /m!. (−1)m M
(5.4)
m=0
n given in Lemmas 3.1 and 3.2, while the For the convergence of An f we shall use the estimates on M convergence of Bn f requires the estimates on Bn,m (x) − Hm (x) given by Proposition 4.2. By Theorem 2.2, ∞
(−1)m G(m) , f Hm (x)/m! → 0 locally uniformly as n → ∞,
m=n
and so we need only to prove that if f ∈ H ν (R) for some ν ≥ 3/2, then An f (x) and Bn f (x) tend to zero locally uniformly for x ∈ R as n → ∞.
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Now, ∞
(m) , f = (−1)m G(m) , f − (−1)m M n
(m) (x) f (x)dx (−1)m G(m) (x) − M n
−∞
∞ (m) (−x) f (x)dx = G(m) (−x) − M n −∞
=
1 2π
∞
n (u) f(u)du, −M (iu)m G(u)
−∞
which gives ∞ m (m) m (m) n (u) f(u) du = Im,n + Jm,n , −M |u|m G(u) (−1) G , f − (−1) Mn , f ≤
(5.5)
−∞
where n
α
|u| G(u) − M n (u) f(u) du, m
Im,n := −nα
n (u) f(u) du, −M |u|m G(u)
Jm,n := |u|>nα
and 0 < α < 1/4. It follows from (5.3) and (5.5) that |An f (x)| ≤
n−1
Im,n
m=0
n−1 |Hm (x)| |Hm (x)| + . Jm,n m! m! m=0
(5.6)
By Lemma 3.1 and as in the proof of (2.10),
Im,n ≤ ≤
K n1−4α
n
α
|u|m e−u
2
/2
|f(u)|du ≤
−nα
⎧∞ ⎨
Kf H ν n1−4α ⎩
u2(m−ν) e−u du 2
0
⎫1/2 ⎬ ⎭
Kf H ν 1/2 Γ (m − ν + 1/2) , n1−4α
(5.7)
for all sufficiently large n, where K is a generic constant independent of m, n. We shall show that n−1
Im,n |Hm (x)|G(x)/m! → 0 uniformly on R as n → ∞.
m=0
By (2.8), |Hm (x)|G(x) ≤ Since
2
m−1 2
π
Γ ((m + 1)/2) .
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Γ ((m + 1)/2) = n−1
Im,n |Hm (x)|G(x)/m! ≤
m=0
j!,
m = 2j + 1,
(2j)! √ 22j j! π,
n−1
17
m = 2j, m−1
Im,n
m=0
1 =√ 2π
2 2 Γ ((m + 1)/2) πm!
[(n−1)/2]
I2j,n
j=0
1 1 + 2j j! π
[(n−2)/2]
I2j+1,n
j=0
2j j! . (2j + 1)!
(5.8)
Now, as H ν (R) ⊂ H 2 (R) for ν > 2, we may assume that 3/2 ≤ ν ≤ 2. By the functional relation Γ(x + 1) = xΓ(x), Γ (m − ν + 1/2) = (m − ν − 1/2)(m − ν − 3/2) · · · (5/2 − ν)Γ(5/2 − ν). So using (5.7) and absorbing Γ(5/2 − ν)1/2 into the generic constant K, for m = 3, 4, . . . , 2 Im,n ≤
K 2 f 2H ν (m − ν − 1/2)(m − ν − 3/2) · · · (5/2 − ν). n2(1−4α)
Then for j = 2, 3, . . . ,
1 j 2 j!
I2j,n
2 ≤
j j K 2 f 2H ν & 2 − ν − 1/2 & 2 − ν − 3/2 2
2
n2(1−4α) =2 =2
=
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & 1 − 1 − 2
2
n2(1−4α) =2 =2
≤
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & exp − exp − 2
2
n2(1−4α) =2 =2
j ν+1 K 2 f 2H ν = 2(1−4α) exp −
n =2
≤
K f 2H ν n2(1−4α) 2
a2 exp(−(ν + 1) ln j) =
K 2 a2 f 2H ν , n2(1−4α) j ν+1
for some positive constant a, independent of j. Thus
[(n−1)/2]
I2j,n
j=2
Kaf H ν 1 ≤ j 2 j! n1−4α
[(n−1)/2]
j=2
1 j
ν+1 2
≤
Kf H ν , n1−4α
where K is a generic constant independent of n. Similarly, for j = 2, 3, . . . , I2j+1,n
2j j! (2j + 1)!
2 ≤
j j K 2 f 2H ν & 2 + 1/2 − ν & 2 − 1/2 − ν 2 + 1 2 + 1 n2(1−4α) =2 =2
=
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & 1 − 1 − 2 + 1 2 + 1 n2(1−4α) =2 =2
(5.9)
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18
j 2ν + 2 K 2 f 2H ν ≤ 2(1−4α) exp − 2 + 1 n =2 ≤
K 2 b2 f 2H ν K 2 b2 f 2 ν exp(−(ν + 1) ln j) = 2(1−4α) H , 2(1−4α) n n j ν+1
for some positive constant b, independent of j. Thus
[(n−2)/2]
Kbf H ν 2j j! ≤ (2j + 1)! n1−4α
I2j+1,n
j=2
[(n−2)/2]
1 j
j=2
≤
ν+1 2
Kf H ν , n1−4α
(5.10)
where K is a generic constant independent of n. By (5.8)–(5.10), n−1
Im,n |Hm (x)|G(x)/m! ≤
m=0
Kf H ν , for all x ∈ R. n1−4α
(5.11)
H Since K f n1−4α converges to 0 as n → ∞, the sequence of partial sums on the left converges uniformly on R. On the other hand, ν
Jm,n ≤ J1,m,n + J2,m,n ,
(5.12)
where
m −u2 /2
|u| e
J1,m,n :=
|f(u)|du,
J2,m,n :=
|u|>nα
|u| M n (u) f(u) du. m
|u|>nα
By the Cauchy–Schwarz inequality, ⎧ ⎪ ⎨
J1,m,n ≤
|u|2(m−ν) e−u du 2
⎪ ⎩
|u|>nα
⎧ ∞ ⎨ =
⎩
tm−ν−1/2 e−t dt
n2α
where f H ν α := n
|u|>nα
|f(u)|2 |u|2ν du
n−1
⎫1/2 ⎧ ⎪ ⎬ ⎪ ⎨ ⎪ ⎭
⎫1/2 ⎬ ⎭
|f(u)|2 |u|2ν du
⎪ ⎩
|u|>nα
⎫1/2 ⎪ ⎬ ⎪ ⎭
f H ν α ≤ Γ(m − ν + 1/2)1/2 f H ν α , n
n
1/2 . As in the proof of (5.11),
J1,m,n |Hm (x)|G(x)/m! ≤ Kf Hnν α , for all x ∈ R.
(5.13)
m=0
By Lemma 3.2,
J2,m,n ≤
⎧ ⎪ ⎨ ⎪ ⎩
|u|>nα mα
≤
n
e
⎫1/2 2 ⎪ ⎬ (m−ν) n (u) du u f Hnν α M ⎪ ⎭
f Hnν α 1 2α 2n
⎧ ⎪ ⎨ ⎪ ⎩
|u|>nα
⎫1/2 ⎪ ⎬
1 du |u|2ν ⎪ ⎭
≤
Knα(m−ν+1/2) f Hnν α 1
e2n
2α
,
(5.14)
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for all sufficiently large n, where K is a generic constant that is independent of m, n. It follows from (2.8) and (5.14) that n−1
J2,m,n |Hm (x)|G(x)/m! ≤
m=0
n−1
Kf Hnν α 1
nα(ν−1/2) e 2 n
nαm (m−1)/2 2 Γ m! m=0
2α
m+1 2
,
and since
2j j!, 2(m−1)/2 Γ ((m + 1)/2) = (2j)! π
m = 2j + 1,
2,
2j j!
m = 2j,
this gives n−1
J2,m,n |Hm (x)|G(x)/m! ≤
m=0
≤ ≤
⎛
Kf
ν Hn α α(ν−1/2) 12 n2α
n
e
1
j=0
[(n−1)/2]
2α
Kf Hnν α nα(ν−1/2) e
⎝ ⎛
Kf Hnν α nα(ν−1/2) e 2 n
[(n−1)/2]
1 2α 2n
⎝
j=0
1
e2n
2α
2jα
n + 2j j!
[(n−2)/2]
1
2α
n
j
2 j! ⎠ (2j + 1)!
j=0
n2jα + nα 2j j!
+ nα e 2 n
⎞ (2j+1)α
[(n−2)/2]
j=0
⎞ n2jα ⎠ 2j j!
.
(5.15)
Hence, by (5.12), (5.13) and (5.15), for all x ∈ R, n−1
Jm,n |Hm (x)|G(x)/m! ≤ 3Kf Hnν α .
(5.16)
m=0
Combining (5.11) and (5.16) gives n−1
(Im,n + Jm,n )|Hm (x)|G(x)/m! ≤
m=0
Kf H ν + 3Kf Hnν α , n1−4α
for all x ∈ R. Since α < 1/4, the sequences on the right converge to 0 as n → ∞, and so the sequence of partial sums on the left converges uniformly on R. By (5.6), An f (x) → 0 locally uniformly as n → ∞. Finally we show that Bn f (x) → 0 locally uniformly on R. By (4.3), n−1 K n−1 m/2 n,m (x) Hm (x) − B m (m) −x2 n(m) , f e e (−1) Mn , f , ≤ (−1)m M m! n m=0 2m m=0 n , f = where K is a generic constant that is independent of m, n. Since (−1)m M (m)
∞ −∞
n (u)f(u)du, (iu)m M
∞ m m (m) (−1) Mn , f ≤ u M n (u) f (u) du −∞ α n m f(u) du + u = (u) M n
−nα
|u|>nα
m n (u) f(u) du, u M
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where 0 < α < 1/2. So n−1 n,m (x) 2 (x) − B H m (m) , f e−x ≤ Tn,1 + Tn,2 , (−1)m M n m!
(5.17)
m=0
where
Tn,1
α n n−1 m K e m/2 n (u) f(u) du, u M := n m=0 2m
(5.18)
−nα
Tn,2
n−1 K e m/2 := n m=0 2m
m n (u) f(u) du. u M
(5.19)
|u|>nα
By (3.1) and as in the derivation of (5.7), α α n n m 2 n (u) f(u) du ≤ u M |u|m e−u /2 f(u) du ≤ Kf H ν Γ(m − ν + 1/2)1/2 ,
−nα
−nα
and so by (5.18), Tn,1 ≤
n−1 e m/2 Kf H ν Γ(m − ν + 1/2)1/2 . n 2m m=0
The series on the right converges as n → ∞ by the ratio test, and so Tn,1 ≤
Kf H ν . n
(5.20)
Also as in the proof of (5.14) and (5.15), (5.19) leads to Tn,2 ≤
1 2α e m/2 K e− 2 n f Hnν α n−1 nmα 1+α(ν−1/2) 2m n m=0
K e− 2 n f Hnν α ≤ n1+α(ν−1/2) 1
2α
(1/2 n−1
n−1 ( e m 1/2 n2mα m! m! 2m m=0 m=0
K e− 2 n f Hnν α 1 n2α ≤ e2 n1+α(ν−1/2) 1
since
n−1 m=0
m!
e m 2m
2α
n−1
e m m! 2m m=0
(1/2 ≤
Kf Hnν α , n1+α(ν−1/2)
(5.21)
converges as n → ∞. Combining (5.17), (5.20) and (5.21) gives
n−1 Kf ν n,m (x) Kf Hnν α 2 H (x) − B m H (m) , f e−x ≤ + 1+α(ν−1/2) (−1)m M , for all x ∈ R. n m! n n m=0
(5.22)
It follows that the sequence of partial sums on the left of (5.22) converges to 0 uniformly on R as n → ∞. Hence, Bn f (x) in (5.4) converges locally uniformly on R. 2
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Proof of Theorem 1.1. For x, t ∈ R, s > 0, let fs,t (x) := f (t − x/s). Then fs,t (u) = sf(−us)e−iust , u ∈ R, and fs,t ∈ H ν (R) if f ∈ H ν (R). As in (5.1)–(5.4), fs,t (x) =
n−1
(m) , fs,t (−1)m M n
m=0
n,m (x) B + Rn fs,t (x), m!
where Rn fs,t (x) =
∞
(−1)m G(m) , fs,t Hm (x)/m! −
m=0
n−1
n,m (x)/m! (m) , fs,t B (−1)m M n
m=0
= An fs,t (x) + Bn fs,t (x) +
∞
(−1)m G(m) , fs,t Hm (x)/m!.
(5.23)
m=n
By (2.12), ∞ ∞ 2 1 m (m) (−1) G , fs,t Hm (x)/m! ≤ Cfs,t H ν ex /2 . (1+ν)/2 m m=n m=n
(5.24)
As in the proof of Theorem 5.1, for α < 1/4, 2
|An fs,t (x)| ≤ K ex
) /2
fs,t H ν + 3fs,t Hnν α n1−4α
* ,
(5.25)
.
(5.26)
and for α < 1/2, |Bn fs,t (x)| ≤ K e
x2
)
fs,t Hnν α fs,t H ν + 1+α(ν−1/2) n n
*
Since
fs,t H ν ≤
1 f H ν , sν−1/2 1/2 s f H ν ,
0 < s < 1, s ≥ 1,
and fs,t Hnν α =
1 sν−1/2
ν f Hsn α,
it follows from (5.23)–(5.26) that Rn fs,t (x) converges to 0 uniformly for t ∈ R and locally uniformly for x ∈ R and s > 0. 2 6. Approximation of derivatives of B-splines by spline framelets In this section we show that the spline framelets constructed by Ron and Shen [3], when suitably standardized, approximate the derivatives of the standardized B-splines and converge to the derivatives of the 1 Gaussian function as the orders of the spline functions tend to infinity with a convergence rate of O( n1−/p ), p 0 < < 1, in the L -norm, 1 ≤ p ≤ ∞. Bernoulli–Schoenberg series expansion formulas of the form (1.5) and (5.1) are then derived with the standardized spline framelets as mother wavelets instead of the derivatives of the B-splines in the wavelet transforms.
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n is The Fourier transform of the standardized B-splines M n (u) = M
n sin(u 3/n) , u 3/n
u ∈ R.
For integers n ≥ 1, m = 0, 1, . . . , n, we define ψ n,m (u) :=
4n 3
= (iu)
m/2 im
m
sin(u
m 3/4n)
n−m n (u/2) cos(u 3/4n) M
n−m 2m sin(u 3/n) sin(u 3/4n) , u 3/n u 3/4n
u ∈ R.
(6.1)
For even n, ψ n,m are, up to constant multiples and a constant dilation, the spline framelets of Ron and Shen [3]. They are also, up to constant multiples and a constant dilation, the mother wavelets for the semidiscrete wavelet transforms arising from the B-spline scale-space considered in [2]. We shall call ψ n,m the n(m) of the standardized standardized Ron–Shen framelets, and show that they approximate the derivatives M B-splines and also converge to the derivatives of the Gaussian function as n → ∞. Theorem 6.1. For any nonnegative integer m,
1 (m) as n → ∞. sup ψ n,m (u) − Mn (u) = O n u∈R
(6.2)
Proof. The Fourier transforms of the derivatives of the standardized B-spline can be expressed as
n sin(u 3/n) u 3/n n−m m m sin u 3/n sin(u 3/4n) m = (iu) . cos(u 3/4n) u 3/n u 3/4n
(m) m M n (u) = (iu)
By (6.1) and (6.3), n−m m sin(u 3/4n) sin(u 3/n) n(m) (u) = um ψ n,m (u) − M Fm,n (u) u 3/n u 3/4n n−m m sin(u 3/n) Fm,n (u), ≤ u u 3/n where sin(u3/4n) m m Fm,n (u) := − cos(u 3/4n) u 3/4n sin(u3/4n) ≤ m − cos(u 3/4n) u 3/4n ∞ 2j Kmu2 2j = m (−1)j−1 , u 3/4n ≤ (2j + 1)! n j=1
(6.3)
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23
for all positive integers n ≥ m and u ∈ R, where K is a constant independent of m, n and u. Hence n−m 3/n) (m) , u ∈ R. Km um+2 sin(u ψ n,m (u) − M n (u) ≤ n u 3/n For |u| ≤
√
(6.4)
n, the proof of (3.1) in Lemma 3.1 gives n−m m+2 sin(u 3/n) u ≤ |u|m+2 e−(n−m)u2 /2n . u 3/n +
The maximum of the expression on the right is attained at u = n > 2(m + 1), and so
n(m+2) n−m ,
(6.5)
which is less than
n−m
(m+2)/2 m+2 sin(u 3/n) m+2 u ≤ e−(m+2)/2 . 1 − m/n u 3/n
√
n for
(6.6)
For |u| ≥ nα , 0 < α < 1/2, similar arguments as in the proof of Lemma 3.2 show that if n > 2(m + 1), n−m m+2 sin(u 3/n) u ≤ nα(m+2) e−n2α /2 em/2n1−2α ≤ Knα(m+2) e−n2α /2 u 3/n
(6.7)
for all sufficiently large n, where K is a generic constant independent of n and u. The estimate (6.2) now follows from (6.4), (6.6) and (6.7). 2 Corollary 6.2. For any nonnegative integer m,
1 m −u2 /2 =O as n → ∞. sup ψ m,n (u) − (iu) e n u∈R Proof. The result follows from Theorems 3.3 and 6.1. 2 Theorem 6.3. For any integer m ≥ 0, (m) ψn,m − M n
∞
=O
1 as n → ∞. n
(6.8)
Proof. By taking inverse Fourier transforms, for any positive integer n ≥ m and x ∈ R, (m) (x) = 1 ψ n,m (x) − M n 2π
∞ e
iux
(m) ψ m,n (u) − Mn (u) du
−∞
and so ∞ 1 (m) (m) ψ m,n (u) − Mn (u) du ψn,m (x) − Mn (x) ≤ π 0 n−m ∞ 3/n) K um+2 sin(u du, ≤ πn u 3/n 0
(6.9)
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24
by (6.4). For a positive α < 1/2, a similar argument as in the proof of Theorem 3.4 using (6.5) gives α α n−m n n m+2 sin(u 3/n) u du ≤ um+2 e−(n−m)u2 /2n du u 3/n
0
0
≤ 2m+2 Γ((m + 3)/2), for n > 2m,
(6.10)
and n−m ∞ m+2 sin(u 3/n) u du ≤ C n(m+3)α e−n2α /2 , u 3/n α
(6.11)
n
where C is a constant independent of n. It follows from (6.9)–(6.11) that (m) (x) ≤ K 2m+2 Γ((m + 3)/2) + CK n(m+3)α e−n2α /2 , ψn,m (x) − M n πn πn which gives (6.8). 2 Corollary 6.4. For any integer m ≥ 0, ψn,m − G(m)
∞
=O
1 as n → ∞. n
(6.12)
Proof. The estimate (6.12) follows from Theorems 3.4 and 6.3. 2 Remark 3. The problem of approximating the spline framelets of Ron and Shen by derivatives of the Gaussian function is also considered recently by Shen and Xu in [5]. However, their results are different from ours. They consider the nonstandardized Gaussian function ,
n 12 12t2 , Gn, (t) := exp − 2n −
π(2n − )42
t ∈ R,
d and show that for large n and = 1, 2, . . . , n, the derivatives dt Gn, (t − jn /2), where jn = 0 or 1 depending on n is even or odd respectively, approximate the original nonstandardized spline generators ψn, uniformly. An error estimate is also given.
We now give the rate of convergence in the Lp -norm. Theorem 6.5. Take any integer m ≥ 0 and 1 ≤ p < ∞. Then for any 0 < < 1, (m) = O ψn,m − M n p
1
n1−/p
as n → ∞.
(6.13)
Proof. By Theorem 6.3, for any positive < 1 and for all positive integers n ≥ m, n n(m) (x) dx ≤ 2n n(m) ψn,m (x) − M ψ n,m − M
−n
for some constant C > 0. Then for all positive integers n ≥ m,
∞
≤
C , n1−
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n n n(m) (x)dx ≤ C . ψ n,m (x)dx − M n1− 0
25
(6.14)
0
Now for even m, ∞
(m) (x)dx M n
∞ =
0
ψ n,m (x)dx =
0
1 2,
m = 0,
0,
m > 0.
For odd m, defining gn,m−1 by gn,m−1 (u) := (iu)
m−1
n−m 2m sin(u 3/n) sin(u 3/4n) , u 3/n u 3/4n
we have, by (6.1), its derivative gn,m−1 = ψ n,m . Hence
∞
ψ n,m (x)dx −
0
∞
(m−1) (0). (m) (x)dx = −gn,m−1 (0) + M M n n
0
The proof of Theorem 6.3 gives (m−1) gn,m−1 − M n
∞
=O
1 , n
and so for odd m, ∞
ψ n,m (x)dx −
0
∞
(m) (x)dx = O M n
1 . n
0
Therefore, for any integer m ≥ 0, ∞ ∞ (m) (x)dx ≤ C . ψ n,m (x)dx − M n n 0
0
It follows from (6.14) and (6.15) that ∞
∞ 1 (m) (x)dx = O ψ n,m (x)dx − M . n n1− n
n
Then similar arguments as in the proof of Theorem 3.5 give
1 (m) , = O ψn,m − M n n1− 1 and hence (6.13). 2
(6.15)
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Corollary 6.6. Take any integer m ≥ 0 and 1 ≤ p < ∞. Then for any 0 < < 1, (m) ψn,m − G = O p
1 n1−/p
as n → ∞.
Proof. The result follows from Theorems 3.5 and 6.5. 2 n(m) of the standardized Since the standardized spline framelets ψ n,m approximate the derivatives M B-splines, it is of interest to investigate to what extent the latter can be replaced by the former in (1.5) and (5.1) of Theorems 1.1 and 5.1 respectively. The next theorem gives a result of this form. Theorem 6.7. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then
f (x) =
√ 0≤m< n
(−1)m ψ n,m , f
n,m (x) B n f (x), +R m!
n f (x) → 0 as n → ∞ locally uniformly for x ∈ R. where R Proof. The proof uses Theorem 5.1 and similar techniques as in its proof as well as the results on the approximation of the derivatives of the standardized B-splines by the standardized spline framelets. If √ f ∈ H ν (R), ν ≥ 3/2, and is continuous, by the proof of Theorem 5.1 (with the sum over 0 ≤ m < n replacing the sum over 0 ≤ m ≤ n − 1), f (x) =
√ 0≤m< n
(m) , f (−1)m M n
n,m (x) B + Rn f (x), m!
n , this gives where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R. Recalling that ψ n,0 = M f (x) =
(−1)m ψ n,m , f
√ 0≤m< n
−
n,m (x) B m!
n(m) , f Bn,m (x) + Rn f (x). (−1)m ψ n,m − M m!
√ 1≤m< n
Let Sn (x) :=
√ 1≤m< n
(m) , f Bn,m (x) , (−1)m+1 ψ n,m − M n m!
n f (x) := Sn (x) + Rn f (x). R
We want to show that Sn (x) → 0 locally uniformly for x ∈ R, as n → ∞. Applying Parseval’s identity,
∞ n,m (x)| 1 |B (m) |Sn (x)| ≤ ψ n,m (u) − Mn (u) f (u)du 2π m! √ 1≤m< n
−∞
≤
1 2π
√ 1≤m< n
⎧ nα ⎨ (m) (u) − M ψ f (u) (u) du n n,m ⎩ −nα
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+ |u|>nα
=
√ 1≤m< n
27
⎫ ⎪ ⎬ (m) f(u) du |Bn,m (x)| ψ n,m (u) − M (u) n ⎪ m! ⎭ (Im,n,1 + Im,n,2 )
n,m (x)| |B , m!
(6.16)
where
Im,n,1
1 := 2π
Im,n,2 :=
and 0 < α < 1/4. For 1 ≤ m <
√
1 2π
α n (m) f(u) du, ψ n,m (u) − M (u) n
−nα
(m) ψ n,m (u) − M f (u) (u) du, n
|u|>nα
n, by Proposition 4.2, B e m/2 2 n,m (x) − Hm (x) K ex ≤ m! n 2m
where K is a generic constant independent of m, n, and so this together with (2.9) give n,m (x)| K e m/2 x2 |Hm (x)| |B ≤ e + m! n 2m m! m/2 2 e K K e m/2 x2 /2 ≤ ex + √ e n 2m m m K e m/2 x2 ≤√ e . m m
(6.17)
Now,
Im,n,1
1 = 2π
α n (m) f(u) du ψ n,m (u) − M (u) n
−nα
⎫ ⎧ nα 2 ⎬1/2 f H ν ⎨ (m) ≤ √ (1 + |u|2 )−ν ψ n,m (u) − M . n (u) du ⎭ 2π ⎩ α −n
By (6.4) and using the same method as in the proof of (6.10),
Im,n,1 ≤
⎧ nα ⎨
Kmf H ν ⎩ n
|u|2m+4−2ν
−nα
≤
Kmf n
Hν
Kmf H ν ≤ n
sin(u 3/n) u 3/n
⎫1/2 ⎬
2(n−m) du
⎧ nα ⎫1/2 ⎨ ⎬ (n−m) 2 u2m+4−2ν e− n u du ⎩ ⎭
0
n n−m
(2m+5−2ν)/4 Γ(m + 5/2 − ν)1/2 .
⎭
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For m <
√
n, the second factor in the last expression is bounded, and so
Kmf H ν Γ(m + 5/2 − ν)1/2 . n √ Also by (6.4) and (6.7), for |u| > nα , m < n and α < 1/4, Im,n,1 ≤
(6.18)
Km α(m+2) −n2α /2 m/2n1−2α (m) ≤ ψ n,m (u) − M n (u) e e n n ≤
K n(1−4α)/2
nmα e−n
2α
/2 1/2n(1−4α)/2
e
K
≤
n(1−4α)/2
nmα e−n
2α
/2
,
and so ⎧ ⎪ ⎨
f H ν Im,n,2 ≤ √ 2π ⎪ ⎩
|u|>nα
≤
⎫1/2 2 ⎪ ⎬ (m) |u|−2ν ψ n,m (u) − M n (u) du ⎪ ⎭ ⎧ ⎪ ⎨
Kf H ν mα −n2α /2 n e ⎪ n(1−4α)/2 ⎩
|u|−2ν du
|u|>nα
≤
⎫1/2 ⎪ ⎬ ⎪ ⎭
Kf H ν α(m−ν+1/2) −n2α /2 n e . n(1−4α)/2
(6.19)
It follows from (6.16)–(6.19) that |Sn (x)| ≤
Kf H ν n +
√ 1≤m< n
Kf H ν n(1−4α)/2
1 e m/2 x2 mΓ(m + 5/2 − ν)1/2 √ e m m
√ 1≤m< n
nα(m−ν+1/2) e−n
2α
/2
1 e m/2 x2 √ e =: Sn,1 (x) + Sn,2 (x). m m
Noting that for ν ≥ 3/2, Γ(m + 5/2 − ν) ≤ Γ(m + 1), and applying the Stirling formula,
m m/2 e m/2 2 ex e m
Sn,1 (x) ≤
Kf H ν n
=
Kf H ν n
≤
Kf H ν x2 e → 0 locally uniformly for x ∈ R. n1/16
√ 1≤m< n
√ 1≤m< n
m3/4
Kf H ν m15/8 x2 e ≤ m9/8 n1/16
√ 1≤m< n
1 x2 e m9/8
Applying exactly the arguments of the proof of (5.15) for the second sum, Kf H ν 1 + nα x2 e n(1−4α)/2 nα(ν−1/2) Kf H ν 2 ≤ (1−4α)/2 ex , n
|Sn,2 (x)| ≤
n f (x) converge locally uniformly to 0 as n → ∞. 2 since ν ≥ 3/2. Therefore Sn,2 (x) and hence Sn (x) and R
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We conclude with the following theorem, which can be derived from the proof of Theorem 6.7 in the same way as Theorem 1.1 from the proof of Theorem 5.1. Theorem 6.8. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then for x, t ∈ R, s > 0, f (x) = SM n f (s, t) +
√ 1≤m< n
n,m (s(t − x)) (−1)m B n f (x, s, t), Wψ n,m f (s, t) + R m!
n f (x, s, t) → 0 as n → ∞ locally uniformly for x ∈ R and s > 0 and uniformly for t ∈ R, and where R SM n,m f (s, t) are continuous wavelet transforms of f n f (s, t) is the B-spline scale-space transform of f , Wψ with the standardized spline framelets as mother wavelets. References [1] R. Brinks, On the convergence of derivatives of B-splines to derivatives of the Gaussian function, Comput. Appl. Math. 27 (2008) 79–92. [2] S.S. Goh, T.N.T. Goodman, S.L. Lee, Singular integrals, scale-space and wavelet transforms, J. Approx. Theory 176 (2013) 68–93. [3] A. Ron, Z. Shen, Affine systems in L2 (Rd ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997) 408–447. [4] I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946) 45–99, 112–141. [5] Z. Shen, Z. Xu, On B-spline framelets derived from the unitary extension principle, SIAM J. Math. Anal. 45 (2013) 127–151. [6] G. Szegö, Orthogonal Polynomials, 4th edition, AMS Colloquium Publications, vol. XXIII, American Mathematical Society, 1975. [7] Y. Xu, R. Wang, Asymptotic properties of B-splines, Eulerian numbers and cube slicing, J. Comput. Appl. Math. 236 (2011) 988–995. [8] J.M. Whittaker, On Lidstone’s series and two-point expansions of analytic functions, Proc. Lond. Math. Soc. 36 (1934) 451–469.
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Contents lists available at ScienceDirect
Applied and Computational Harmonic Analysis www.elsevier.com/locate/acha
Appell sequences, continuous wavelet transforms and series expansions Say Song Goh a , Tim N.T. Goodman b , S.L. Lee a,∗ a
Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260, Singapore b Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, The University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
a r t i c l e
i n f o
Article history: Received 19 March 2013 Received in revised form 23 August 2015 Accepted 15 January 2016 Available online xxxx Communicated by Ding-Xuan Zhou Keywords: Appell sequences Scale-space Wavelet transforms Gaussian Hermite polynomials B-splines
a b s t r a c t A series expansion with remainder for functions in a Sobolev space is derived in terms of the classical Bernoulli polynomials, the B-spline scale-space and the continuous wavelet transforms with the derivatives of the standardized B-splines as mother wavelets. In the limit as their orders tend to infinity, the B-splines and their derivatives converge to the Gaussian function and its derivatives respectively, the associated Bernoulli polynomials converge to the Hermite polynomials, and the corresponding series expansion is an expansion in terms of the Hermite polynomials, the Gaussian scale-space and the continuous wavelet transforms with the derivatives of the Gaussian function as mother wavelets. A similar expansion is also derived in terms of continuous wavelet transforms in which the mother wavelets are the spline framelets that approximate the derivatives of the standardized B-splines. © 2016 Published by Elsevier Inc.
1. Introduction The object of this paper is to present some new results involving the classical Bernoulli and Hermite polynomials, and to connect scale-space and wavelet transforms with classical series expansions of realvalued functions in terms of these Appell polynomials. A brief review and a general approach to scale-space and continuous wavelet transforms in the context of singular integral operators can be found in [2]. The Bernoulli polynomials are associated with the uniform B-splines in the same way as the Hermite polynomials are associated with the Gaussian function. The main result is Theorem 1.1 below, but in order to give a systematic presentation, we shall begin with the general setting and consider a compactly supported probability measure or probability density function μ, of which the standardized uniform B-splines are examples. Then for any integer m ≥ 0, * Corresponding author. E-mail addresses: [email protected] (S.S. Goh), [email protected] (T.N.T. Goodman), [email protected] (S.L. Lee). http://dx.doi.org/10.1016/j.acha.2016.01.005 1063-5203/© 2016 Published by Elsevier Inc.
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μ(m) , e(·)z = (−1)m μ, z m e(·)z = (−1)m z m μ (iz) and so m (m)
(−1) μ
e(·)z , μ (iz)
= zm
(1.1)
in a neighborhood of 0, where the derivatives μ(m) are in the sense of distribution, ·, · denotes the action ∞ of distributions on test functions, and μ is the Fourier transform defined by μ (u) := −∞ e−iut dμ(t). Since μ is compactly supported, μ is analytic, and so we can define a sequence of Appell polynomials Pμ,m by the generating function ∞ exz Pμ,m (x) m = z , μ (iz) m=0 m!
which, together with (1.1), gives the biorthogonal relation Pμ,m = δm,n . (−1)n μ(n) , m!
(1.2)
∞ The biorthogonal system {(−1)m μ(m) }∞ m=0 , {Pμ,m /m!}m=0 provides a formal biorthogonal expansion for ∞ f ∈ C of the form
f (x) =
∞
(−1)m μ(m) , f
m=0
∞ Pμ,m (x) Pμ,m (x) = . μ, f (m) m! m! m=0
(1.3)
Such expansions include a large class of classical series expansions, such as Taylor’s series, the Euler– Maclaurin expansion and the Lidstone series [8]. For the case where μ is the Gaussian density function 2 G(t) := √12π e−t /2 , which is not compactly supported but the above procedure is applicable, the corresponding Appell sequence comprises the Hermite polynomials, the biorthogonal relation (1.2) is the orthonormal relation with respect to the weighted inner product with Gaussian weight and the expansion (1.3) is the classical Hermite polynomial expansion. This paper focuses on the case where the probability density functions are the standardized B-splines and derives an expansion of the form (1.3) for functions in a Sobolev space. To this end, we define for ν > 1, the Sobolev space H ν (R) that comprises all tempered distributions f for which f 2H ν
1 := 2π
∞
|f(u)|2 (1 + |u|2 )ν du < ∞.
−∞
For the forward uniform B-splines Mn of order n, the Fourier transforms are n (u) = M
1 − e−iu iu
n ,
u ∈ R,
and so n (iz) = M
ez − 1 z
n ,
z ∈ C.
The corresponding Appell sequences of polynomials, which we denote by Bn,m , are the Bernoulli polynomials of order n generated by
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3
|z| < 2π.
n As a probability distribution, the mean and variance of Mn are μn = n2 and σn2 = 12 respectively. Let n (x) := σn Mn (σn x + μn ) be the standardized B-splines with mean 0 and variance 1. Then M
n (u) = eiun/(2σn ) M
1 − e−iu/σn iu/σn
n u ∈ R.
,
Further the standardized Bernoulli polynomials n,m (x) := 1 Bn,m (σn x + μn ), B σnm are generated by the generating function ∞ exz Bn,m (x) m z , = m! n (iz) m=0 M
|z| < 2πσn ,
(1.4)
where for every x ∈ R, the convergence is uniform on |z| ≤ ρ, for any ρ < 2πσn . The main result is Theorem 1.1. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then for x, t ∈ R, s > 0, f (x) = SM n f (s, t) +
n−1
n,m (s(t − x)) (−1)m B WM (m) f (s, t) + Rn f (x, s, t), n m! m=1
(1.5)
where Rn f (x, s, t) → 0 as n → ∞ locally uniformly for x ∈ R and s > 0 and uniformly for t ∈ R, and ∞ SM n f (s, t) :=
n (s(t − x))f (x)dx, sM
t ∈ R, s > 0,
−∞
is the B-spline scale-space transform of f , ∞ WM (m) f (s, t) := n
(m) (s(t − x))f (x)dx, sM n
t ∈ R, s > 0,
−∞
are continuous wavelet transforms of f with the derivatives of the standardized B-splines as mother wavelets. The representation of f (x) in (1.5) holds for any t ∈ R and s > 0. In particular, if t = 0 and s = 1, it becomes f (x) =
n−1
(m) , f (−1)m M n
m=0
n,m (x) B + Rn f (x), m!
where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R, which is the expansion of f , with remainder, in n,m /m!}. On the other hand, if t = x, then (1.5) gives n(m) }, {B terms of the biorthogonal system {(−1)m M f (x) = SM n f (s, x) +
n−1
n,m (0) (−1)m B WM (m) f (s, x) + Rn f (s, x), n m! m=1
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which is a decomposition of f into its B-spline scale-space and wavelet transforms. Since spline functions were introduced by Schoenberg [4], such an expansion involving spline functions and Bernoulli polynomials will be referred as the Bernoulli–Schoenberg series. To prove Theorem 1.1, we first develop the corresponding expansion where the Gaussian function G(t) = 2 √1 e−t /2 defines the Gaussian scale-space 2π ∞ sG(s(t − x))f (x)dx,
SG f (s, t) :=
t ∈ R, s > 0,
−∞
and its derivatives G(m) , m = 1, 2, . . . , are mother wavelets that define the continuous wavelet transforms ∞ sG(m) (s(t − x))f (x)dx,
WG(m) f (s, t) :=
t ∈ R, s > 0.
−∞
The Gaussian function also defines the Appell sequence of Hermite polynomials Hm , m = 0, 1, . . . , by the generating function ∞ exz Hm (x) m z , = G(iz) m=0 m!
(1.6)
2 where G(u) = e−u /2 is the Fourier transform of G, and for every x ∈ R, the series converges locally uniformly on C. Also (−1)m G(m) (t) = Hm (t)G(t) and Hm /m! are biorthogonal sequences, which is the well-known √ fact that the normalized Hermite polynomials Hm / m!, m = 0, 1, . . . , form an orthonormal basis for the ∞ Hilbert space L2G (R) with inner product f, gG := −∞ f (t)g(t)G(t)dt, and so for every f ∈ L2G (R),
∞
f (x) =
f, Hm G
m=0
Hm (x) , m!
where the convergence is in L2G (R). We shall show that if f ∈ H ν (R) for some ν > 1 and is continuous, then f (x) = SG f (s, t) +
∞ Hm (s(x − t)) WG(m) f (s, t), m! m=1
(1.7)
where the series converges uniformly in t ∈ R and locally uniformly in x ∈ R and s > 0. Note that if t = 0 and s = 1, the resulting series is the Hermite expansion in Sobolev space. On the other hand, since
Hm (0) =
0, (−1)j (2j)! , 2j j!
m = 2j + 1, m = 2j,
j = 0, 1, . . . ,
if t = x, the expansion becomes
f (x) = SG f (s, x) +
∞ (−1)j j=1
2j j!
WG(2j) f (s, x),
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5
which is a decomposition of f into its Gaussian scale-space and wavelet transforms. Therefore, (1.7) provides an expansion that unifies the classical Hermite polynomial expansion on one hand and Gaussian scale-space and wavelet decomposition on the other. Section 2 gives a proof of (1.7). The proof of Theorem 1.1 is based on equation (1.7) and requires precise estimates of the rate of convergence of the Fourier transforms of the derivatives of the standardized B-splines to the Fourier transforms of the derivatives of the Gaussian function, as well as on the rate of convergence of the standardized Bernoulli polynomials to the Hermite polynomials as the order of the B-spline tends to infinity. Section 3 studies the rate of convergence of the derivatives of the standardized B-splines, where we also show that for any nonnegative integer m and 0 < < 1, (m) (m) M − G = O n p
1
n1−/p
as n → ∞,
1 ≤ p ≤ ∞,
(m) a result which is of interest in itself. In particular, M − G(m) = O n1 . The convergence of the n ∞ derivatives of B-splines has been established in [1] and pointwise convergence of order O(1/n) is given in [7], where they also deduce Lp -convergence for p ∈ [2, ∞), but no convergence rate in uniform or Lp -norm is given. The rate of convergence of the standardized Bernoulli polynomials to the Hermite polynomials is given in Section 4 and the proof of Theorem 1.1 in Section 5. In Section 6 we show that when suitably standardized, the spline framelets of Ron and Shen [3] approximate the derivatives of the standardized B-splines and converge to the derivatives of the Gaussian function with the same rate as that of the derivatives of the B-splines. Bernoulli–Schoenberg series expansion formulas of the form (1.5) are then derived with the standardized spline framelets as mother wavelets in place of the derivatives of the B-splines in the wavelet transforms. 2. Hermite polynomial expansion, Gaussian scale-space and wavelet transforms The Gaussian function G(t) := √12π e−t /2 defines the Gaussian scale-space, and its derivatives G(m) , m = 1, 2, . . . , are mother wavelets that define continuous wavelet transforms. It also generates the Appell sequence of Hermite polynomials, Hm , m = 0, 1, . . . , by the generating function (1.6). Let L2G (R) be the ∞ Hilbert space with inner product f, gG := −∞ f gG. It is well known that G(m) = (−1)m Hm G, m = 0, 1, . . . , and 2
Hj , Hm /m!G = δjm ,
j, m = 0, 1, . . . .
Any f ∈ L2G (R) can be represented by
f=
∞
f, Hm G Hm /m!
(2.1)
m=0
in L2G (R). Letting f, g denote the usual L2 inner product, for any f ∈ L2 (R) ⊂ L2G (R), (2.1) implies that
f=
∞
(−1)m G(m) , f Hm /m!
m=0
in L2G (R). For t, x ∈ R and s > 0, let fs,t (x) := f (t − x/s). Then for m = 0, 1, . . . ,
(2.2)
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∞ (−1) G m
(m)
, fs,t = (−1)
G(m) (x)f (t − x/s)dx
m −∞
∞ m
= (−1)
sG(m) (s(t − y))f (y)dy,
−∞
and so G, fs,t = SG f (s, t)
(2.3)
(−1)m G(m) , fs,t = (−1)m WG(m) f (s, t).
(2.4)
and for m = 1, 2, . . . ,
Therefore, we have by (2.2),
fs,t = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm /m!
m=1
in L2G (R) for any t ∈ R and s > 0. To obtain uniform convergence we require f to belong to some Sobolev space. Here we are concerned with H ν (R) for some ν > 1, and in this case, H ν (R) ⊂ L2 (R). We shall prove Theorem 2.1. If f ∈ H ν (R) for some ν > 1 and is continuous, then for any t, x ∈ R and s > 0, f (t − x/s) = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm (x)/m!,
(2.5)
m=1
where the convergence is uniform over t ∈ R and locally uniform over s > 0 and x ∈ R. Remark 1. Since Hm (t) = (−1)m Hm (−t), a change of variable in (2.5) gives the following equivalent representation
f (x) = SG f (s, t) +
∞ Hm (s(x − t)) WG(m) f (s, t). m! m=1
First we show that under the conditions of Theorem 2.1, convergence is locally uniform in (2.1). The method and estimates in the proof of Theorem 2.2 below will also be used elsewhere in the paper. Theorem 2.2. If f ∈ H ν (R) for some ν > 1 and is continuous, then
f (x) =
∞
(−1)m G(m) , f Hm (x)/m!,
m=0
where the convergence is locally uniform over x ∈ R.
(2.6)
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7
Proof. For n = 0, 1, . . . , ∞
2
n
f (x)G(x) −
(−1) G m
(m)
m=0
−∞
∞
n
f (x) −
≤
(−1) G m
m=0
−∞
(m)
Hm (x) G(x) , f m!
Hm (x) , f m!
dx
2 G(x)dx
2 n m (m) = f − (−1) G , f Hm /m! . m=0
G
Now Hm G ∈ L2 (R), and since H ν (R) ⊂ L2 (R), f G also belongs to L2 (R), and by (2.2), this gives f (x)G(x) =
∞
(−1)m G(m) , f Hm (x)G(x)/m!
(2.7)
m=0
in L2 (R). We want to show that the series on the right of (2.7) converges uniformly. For m = 0, 1, . . . , x ∈ R, m
(−1) Hm (x)G(x) = G
(m)
1 (x) = 2π
∞
(iu)m e−u
2
/2 iux
e
du,
−∞
and so 1 |Hm (x)G(x)| ≤ π
∞
um e−u
2
/2
du =
2
m−1 2
π
Γ ((m + 1)/2) .
(2.8)
0
By the Stirling formula, √
Γ ((m + 1)/2) ∼
2π m m/2 , 2m/2 e
m! = Γ(m + 1) ∼
m m √ 2πm , e
and so for all sufficiently large m, 1 e m/2 |Hm (x)G(x)| ≤ K√ . m! m m
(2.9)
Also for m = 0, 1, . . . , ∞ (−1) G m
(m)
, f = (−1)
m
(m)
f (t)G
−∞
(−1)m (t)dt = 2π
∞
2 f(u)(−iu)m e−u /2 du
−∞
and so for some constant C > 0,
|(−1) G m
(m)
1 , f | ≤ 2π
∞ −∞
|f(u)||u|ν |u|m−ν e−u
2
/2
du
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8
⎛ ≤ Cf H ν ⎝
∞
⎞1/2 |u|2m−2ν e−u du⎠ 2
−∞
⎛
= Cf H ν ⎝2
∞
⎞1/2 u2m−2ν e−u du⎠ 2
0 1/2
= Cf H ν Γ (m − ν + 1/2)
.
(2.10)
Again by the Stirling formula, Γ (m − ν + 1/2) ∼
2π(m − ν − 1/2)
m − ν − 1/2 e
m−ν−1/2 ∼
√
2π
1 m m , mν e
and so (2.10) gives |(−1)m G(m) , f | ≤ C
f H ν m m/2 , e mν/2
(2.11)
where the constant C is a generic constant independent of m. It follows from (2.9) and (2.11) that f H ν (−1)m G(m) , f Hm (x)G(x)/m! ≤ C (1+ν)/2 , m
(2.12)
for all x ∈ R. Since ν > 1, we conclude that the series in (2.7) converges uniformly on R. It follows that (2.6) holds locally uniformly on R. 2 Remark 2. Although Theorem 2.1 involves classical Hermite polynomial expansion, we are unable to find similar results in the literature. More information on local uniform convergence of Hermite polynomial expansions can be found in the classic monograph of Szegö ([6], Theorem 9.1.6). Proof of Theorem 2.1. For any t ∈ R and s > 0, let fs,t (x) := f (t − x/s), x ∈ R. Then fs,t ∈ H ν (R) and its Fourier transform is given by fs,t (u) = sf(−us)e−iust , u ∈ R. By Theorem 2.2, for t ∈ R and s > 0, fs,t (x) =
∞
(−1)m G(m) , fs,t Hm (x)/m!,
(2.13)
m=0
where the convergence is locally uniform over x ∈ R. A calculation similar to (2.10) shows that for some constant C > 0, (−1)m G(m) , fs,t ≤ C
1 sν−1/2
f H ν Γ(m − ν + 1/2)1/2 ,
t ∈ R, s > 0.
Proceeding as in the proof of Theorem 2.2, we see that the convergence of the series in (2.13) is uniform over t ∈ R and locally uniform over s > 0. Combining with (2.3) and (2.4) gives (2.5). 2 Corollary 2.3. If for some ν > 1, f ∈ H ν (R) and is continuous, then f (t) = SG f (s, t) +
∞ (−1)j j=1
2j j!
WG(2j) f (s, t),
where the convergence is uniform over t ∈ R and locally uniform over s > 0.
(2.14)
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Proof. Since f is continuous, we may set x = 0 in (2.5) to give f (t) = SG f (s, t) +
∞
(−1)m WG(m) f (s, t)Hm (0)/m!,
(2.15)
m=1
where the convergence is uniform over t ∈ R and locally uniform over s > 0. Now 1 G(m) (0) = (−1)m Hm (0)G(0) = √ (−1)m Hm (0), 2π and since G(m) (0) = 0 if m is odd and G(2j) (0) =
(−1)j (2j)! √1 , 2j j! 2π
(2.14) follows from (2.15). 2
3. Convergence of derivatives of B-splines We shall first prove some lemmas, which will be used to derive the rate of convergence of the derivatives of the standardized B-splines in the frequency domain as well as in the proof of the main theorem. Lemma 3.1. For |u| ≤ nα , 0 < α < 1/2, n (u) ≤ e−u2 /2 0≤M
(3.1)
and for all sufficiently large n, 4 −u2 /2 n (u) − e−u2 /2 ≤ K u e M , n where K is an absolute constant. Proof. By Taylor’s theorem, for |u| ≤ n (u) = M
n/3 π,
n
n sin u 3/n 3u4 u2 + ≤ 1− . 2n 40n2 u 3/n
Now for 0 ≤ s ≤ 6/5, e−s ≥ 1 − s + s2 /2 − s3 /6 ≥ 1 − s + 3s2 /10, and so for u2 ≤ 12n/5, 1−
2 3u4 u2 + ≤ e−u /2n , 2n 40n2
which gives (3.1). 6 u2 u4 u n (u) = e− 2 − 20n +O n2 , and so Also M 4 6 6 u4 u n (u) − e−u2 /2 = e−u2 /2 1 − e− 20n +O n2 ≤ e−u2 /2 u + O u M 20n n2 2 2 2 e−u /2 |u|4 1 |u|4 e−u /2 u = 20 + O n ≤ K n n for |u| ≤
√
n for all sufficiently large n.
2
(3.2)
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Lemma 3.2. Take integers m > 0, n > m and 0 < α < 1/2. Then for all sufficiently large n, m n (u) ≤ nmα e− 12 n2α for u ≥ nα . u M Proof. Let Kn,m
Kn,m
(3.3)
m := supu≥nα u M n (u). Then
n n m/2 n m/2 | sin x|n m sin u 3/n = sup u = Ann,m , = √sup n(1−m/n) x≥ 3nα−1/2 3 3 x u 3/n u≥nα
where An,m :=
sup
√ x≥ 3nα−1/2
| sin x| . x1−m/n
Observe that | sin x| sin x = sup , 1−m/n 1−m/n x≥π/2 x π/2≤x≤π x sup
and so for
√ α−1/2 3n ≤ π/2, An,m = √
Consider the function g(x) :=
sin x , x1−m/n
sin x
sup 3nα−1/2 ≤x≤π
x1−m/n
.
0 ≤ x ≤ π, where g(0) = g(π) = 0 and
g (x) = {x cos x − (1 − m/n) sin x}/x2−m/n = cos x{x − (1 − m/n) tan x}/x2−m/n . Then g (x) = 0 only at the unique value x = ζ, 0 < ζ < π/2, where tan ζ = ζ/(1 − m/n). We want to show √ that ζ < 3nα−1/2 . Consider m tan x − x 1 m m − 1− . x− 1− tan x = 3 2 x n nx n x3 At x =
√
3nα−1/2 , as n → ∞, x → 0 and nx2 = 3n2α → ∞. Thus
1 1 tan x − x m =− . x − 1 − tan x = − lim 3 3 n→∞ x x→0 n x 3 √ √ Therefore for all large enough n, g ( 3nα−1/2 ) < 0. Since g (π) < 0, ζ < 3nα−1/2 . Hence g is decreasing √ on [ 3nα−1/2 , π]. Thus lim
An,m = √
sin x
sup 3nα−1/2 ≤x≤π
x1−m/n
√ √ α−1/2 sin( 3nα−1/2 ) ) m/2n m(α−1/2)/n sin( 3n √ = √ =3 n , α−1/2 1−m/n α−1/2 ( 3n ) 3n
and so Kn,m =
n m/2 3
Ann,m
mα
=n
n sin(nα 3/n) . nα 3/n
(3.4)
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11
By (3.1) of Lemma 3.1 with u = nα ,
n sin(nα 3/n) 1 2α ≤ e− 2 n . nα 3/n
It follows from (3.4) that Kn,m ≤ nmα e− 2 n 1
2α
for all large enough n. 2
Theorem 3.3. For any integer m ≥ 0,
m 1 m −u2 /2 =O as n → ∞. sup u M n (u) − u e n u∈R
(3.5)
Proof. For any nonnegative integer m, (3.2) gives m+4 −u2 /2 m e n (u) − um e−u2 /2 ≤ K |u| u M , n
|u| < nα , 0 < α < 1/2.
The right hand side of the inequality attains its maximum at |u| = It follows that for all |u| < nα ,
√
m + 4 < nα for all sufficiently large n.
(m+4)/2
m 1 n (u) − um e−u2 /2 ≤ K m + 4 u M . =O n e n Since |u|m e−u
2
/2
(3.6)
is decreasing for all large enough |u|, we see that for all large enough n
1 , n
|u| ≥ nα .
mα m n (u) ≤ n2α = O 1 , u M en /2 n
|u| ≥ nα .
|um e−u
2
/2
|≤
nmα =O en2α /2
(3.7)
Also from (3.3), for all sufficiently large n,
Therefore,
m n (u) − um e−u2 /2 ≤ um M n (u) + |um e−u2 /2 | = O 1 , u M n
|u| ≥ nα .
By (3.6) and (3.8), we obtain (3.5). 2 Theorem 3.4. For any integer m ≥ 0, (m) Mn − G(m)
∞
1 as n → ∞. =O n
Proof. By taking inverse Fourier transforms, for n ≥ 1 and x ∈ R, (m) (x) − G(m) (x) = 1 M n 2π and so
∞ e −∞
iux
(iu)
m
−u2 /2 du M n (u) − e
(3.8)
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12
∞ 1 (m) (m) n (u) − e−u2 /2 du. um M Mn (x) − G (x) ≤ π
(3.9)
0
By (3.2), for 0 < α < 1/2, n
α n 2 2 K −u /2 n (u) − e du ≤ u M um+4 e−u /2 du n
α
m
0
0
K ≤ n
∞ u
m+4 −u2 /2
e
1 . du = O n
(3.10)
0
Take any integer > 1. Then by (3.3) of Lemma 3.2 with m replaced by m + , for sufficiently large n, (m+)α m n (u) ≤ n1 u M e 2 n2α u ,
u ≥ nα , 0 < α < 1/2.
Also, as in the proof of (3.7), |um e−u
2
/2
|≤
n(m+)α , 1 2α e 2 n u
u ≥ nα ,
and so ∞
∞ ∞ (m+)α m m −u2 /2 2 n −u /2 n (u) − e n (u) + u e u M du ≤ u M du du ≤ 2 1 2α e 2 n u m
nα
nα
nα
2n(m+)α 2n(m+1)α = = 1 2α 1 2α = O ( − 1)e 2 n n(−1)α ( − 1)e 2 n
1 . n
(3.11)
The result follows from (3.9), (3.10) and (3.11). 2 Theorem 3.5. Take any integer m ≥ 0. Then for any 0 < < 1 and 1 ≤ p < ∞, (m) (m) Mn − G = O
1 n1−/p
p
as n → ∞.
(3.12)
Proof. By Theorem 3.4, we have (m) Mn − G(m)
∞
=O
1 . n
(3.13)
Then for any positive < 1 and for all n ≥ 1, n (m) (m) − G(m) Mn (x) − G(m) (x) dx ≤ 2n M n
−n
∞
≤
C , n1−
(3.14)
where C > 0 is a generic constant. Thus for all n ≥ 1, n n (m) (x)dx − G(m) (x)dx ≤ C . M n n1− 0
0
(3.15)
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Now for m even,
∞ 0
∞
n(m) (x)dx = M
(m) (x)dx − M n
0
∞ 0
G(m) (x)dx, and for m odd, by Theorem 3.4,
∞ G
13
(m)
(m−1) (0) + G(m−1) (0) = O (x)dx = −M n
1 , n
0
and so ∞ ∞ (m) (x)dx − G(m) (x)dx ≤ C . M n n 0
(3.16)
0
It follows from (3.15) and (3.16) that ∞
∞ 1 (m) (x)dx − G(m) (x)dx = O M . n n1− n
(3.17)
n
n(m) has exactly m simple Now G(m) has exactly m simple zeros, a1 < a2 < · · · < am , and similarly M n n n n zeros, a1 < a2 < · · · < am . Then from (3.13) limn→∞ aj = aj , j = 1, 2, . . . , m. So for all large enough n, n(m) have no zero on [n , ∞), and therefore G(m) and M ∞
∞ 1 (m) G (x) dx = G(m) (x)dx = O n
n
(3.18)
n
and ∞ ∞
∞ 1 (m) (m) (m) , Mn (x) dx = Mn (x)dx = G (x)dx + O n1−
n
n
(3.19)
n
by (3.17). Then (3.14), (3.18) and (3.19) give
1 (m) . Mn − G(m) = O n1− 1 Now, using the inequality 1/p
1−1/p f p ≤ f ∞ f 1
for any f ∈ L1 (R) ∩ L∞ (R) and 1 < p < ∞, we have (3.12). 2 4. Convergence of Bernoulli polynomials to Hermite polynomials n,m to the Hermite polyTo obtain the rate of convergence of the standardized Bernoulli polynomials B nomials Hm as n → ∞, we note from (1.4) and (1.6) that ∞ exz exz Bn,m (x) − Hm (x) m z , − z2 /2 = m! n (iz) e m=0 M
and so by Cauchy’s integral formula,
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⎛ ⎞ 2 1 Bn,m (x) − Hm (x) 1 dz = exz ⎝ − e−z /2 ⎠ m+1 , m! 2πi z n (iz) M C where C is a circle with center at the origin and radius r < 2πσn = π
(4.1)
n/3.
Lemma 4.1. For z in a compact subset of C,
1 2 1 −z /2 as n → ∞. −e =O n M n (iz) Proof. Take n large enough so that the open disk with center at the origin and radius π given compact set. Then for z in the compact set, a direct computation gives ⎛ Log ⎝
⎞ 1 n (iz) M
2 4 ⎠+ z = z +O 2 20n
z6 n2
(4.2) n/3 contains the
,
and so ⎛ ⎞ 2 1 z Log ⎝ ⎠ + ≤ A. 2 n n (iz) M Therefore, ⎛ ⎞
2 n (iz) 1 Log 1/ M 2 2 1 −z /2 −z /2 Log ⎝ ⎠ + z ≤ KA , e = ≤ K − e − e 2 n M n (iz) n (iz) M where A, K are constants independent of n. 2 Proposition 4.2. For positive integers m < n and x ∈ R, 2 B e m/2 n,m (x) − Hm (x) Kex , ≤ m! n 2m
(4.3)
where K is a generic constant that is independent of m, n. Proof. By (4.1) and (4.2) followed by writing z = ζ + iη, ζ, η ∈ R, ⎛ ⎞ B 2 2 2 1 (x) − H (x) 1 dz n,m −x m xz−x ⎝ −z /2 ⎠ = e −e e m+1 m! z 2πi M n (iz) C xz−x2 xζ−x2 K K e e ≤ |dz| |dz| = 2πn z m+1 2πn rm+1 C
K ≤ 2πnrm+1
C
e−(x−ζ/2) eζ 2
2
/4
|dz|
C
≤
K 2πnrm+1
2
eζ C
2
/4
|dz| ≤
Ker /4 , nrm
(4.4)
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15
r 2 /4 for all positive r < π n/3, where r is the radius of the circle C. The minimum of e rm is attained at √ √ r = 2m, which is less than π n/3, since m < n. Therefore, (4.4) holds in particular for r = 2m, i.e. B K e m/2 n,m (x) − Hm (x) −x2 ≤ , e m! n 2m and so the result follows. 2 5. Proof of Bernoulli–Schoenberg series formula We first prove the following Theorem 5.1. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then
f (x) =
n−1
(m) , f (−1)m M n
m=0
n,m (x) B + Rn f (x), m!
(5.1)
where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R. Proof. We want to show that Rn f (x) := f (x) − uniformly for x ∈ R. By (2.6) of Theorem 2.2,
Rn f (x) =
∞
n−1
m (m) m=0 (−1) Mn , f Bn,m (x)/m!
(−1)m G(m) , f Hm (x)/m! −
m=0
n−1
→ 0 as n → ∞ locally
(m) , f B n,m (x)/m! (−1)m M n
m=0
= An f (x) + Bn f (x) +
∞
(−1)m G(m) , f Hm (x)/m!,
(5.2)
m=n
where
An f (x) :=
n−1
n(m) , f Hm (x)/m!, (−1)m G(m) , f − (−1)m M
(5.3)
m=0
Bn f (x) :=
n−1
n(m) , f Hm (x) − B n,m (x) /m!. (−1)m M
(5.4)
m=0
n given in Lemmas 3.1 and 3.2, while the For the convergence of An f we shall use the estimates on M convergence of Bn f requires the estimates on Bn,m (x) − Hm (x) given by Proposition 4.2. By Theorem 2.2, ∞
(−1)m G(m) , f Hm (x)/m! → 0 locally uniformly as n → ∞,
m=n
and so we need only to prove that if f ∈ H ν (R) for some ν ≥ 3/2, then An f (x) and Bn f (x) tend to zero locally uniformly for x ∈ R as n → ∞.
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16
Now, ∞
(m) , f = (−1)m G(m) , f − (−1)m M n
(m) (x) f (x)dx (−1)m G(m) (x) − M n
−∞
∞ (m) (−x) f (x)dx = G(m) (−x) − M n −∞
=
1 2π
∞
n (u) f(u)du, −M (iu)m G(u)
−∞
which gives ∞ m (m) m (m) n (u) f(u) du = Im,n + Jm,n , −M |u|m G(u) (−1) G , f − (−1) Mn , f ≤
(5.5)
−∞
where n
α
|u| G(u) − M n (u) f(u) du, m
Im,n := −nα
n (u) f(u) du, −M |u|m G(u)
Jm,n := |u|>nα
and 0 < α < 1/4. It follows from (5.3) and (5.5) that |An f (x)| ≤
n−1
Im,n
m=0
n−1 |Hm (x)| |Hm (x)| + . Jm,n m! m! m=0
(5.6)
By Lemma 3.1 and as in the proof of (2.10),
Im,n ≤ ≤
K n1−4α
n
α
|u|m e−u
2
/2
|f(u)|du ≤
−nα
⎧∞ ⎨
Kf H ν n1−4α ⎩
u2(m−ν) e−u du 2
0
⎫1/2 ⎬ ⎭
Kf H ν 1/2 Γ (m − ν + 1/2) , n1−4α
(5.7)
for all sufficiently large n, where K is a generic constant independent of m, n. We shall show that n−1
Im,n |Hm (x)|G(x)/m! → 0 uniformly on R as n → ∞.
m=0
By (2.8), |Hm (x)|G(x) ≤ Since
2
m−1 2
π
Γ ((m + 1)/2) .
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Γ ((m + 1)/2) = n−1
Im,n |Hm (x)|G(x)/m! ≤
m=0
j!,
m = 2j + 1,
(2j)! √ 22j j! π,
n−1
17
m = 2j, m−1
Im,n
m=0
1 =√ 2π
2 2 Γ ((m + 1)/2) πm!
[(n−1)/2]
I2j,n
j=0
1 1 + 2j j! π
[(n−2)/2]
I2j+1,n
j=0
2j j! . (2j + 1)!
(5.8)
Now, as H ν (R) ⊂ H 2 (R) for ν > 2, we may assume that 3/2 ≤ ν ≤ 2. By the functional relation Γ(x + 1) = xΓ(x), Γ (m − ν + 1/2) = (m − ν − 1/2)(m − ν − 3/2) · · · (5/2 − ν)Γ(5/2 − ν). So using (5.7) and absorbing Γ(5/2 − ν)1/2 into the generic constant K, for m = 3, 4, . . . , 2 Im,n ≤
K 2 f 2H ν (m − ν − 1/2)(m − ν − 3/2) · · · (5/2 − ν). n2(1−4α)
Then for j = 2, 3, . . . ,
1 j 2 j!
I2j,n
2 ≤
j j K 2 f 2H ν & 2 − ν − 1/2 & 2 − ν − 3/2 2
2
n2(1−4α) =2 =2
=
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & 1 − 1 − 2
2
n2(1−4α) =2 =2
≤
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & exp − exp − 2
2
n2(1−4α) =2 =2
j ν+1 K 2 f 2H ν = 2(1−4α) exp −
n =2
≤
K f 2H ν n2(1−4α) 2
a2 exp(−(ν + 1) ln j) =
K 2 a2 f 2H ν , n2(1−4α) j ν+1
for some positive constant a, independent of j. Thus
[(n−1)/2]
I2j,n
j=2
Kaf H ν 1 ≤ j 2 j! n1−4α
[(n−1)/2]
j=2
1 j
ν+1 2
≤
Kf H ν , n1−4α
where K is a generic constant independent of n. Similarly, for j = 2, 3, . . . , I2j+1,n
2j j! (2j + 1)!
2 ≤
j j K 2 f 2H ν & 2 + 1/2 − ν & 2 − 1/2 − ν 2 + 1 2 + 1 n2(1−4α) =2 =2
=
j
j ν + 1/2 & ν + 3/2 K 2 f 2H ν & 1 − 1 − 2 + 1 2 + 1 n2(1−4α) =2 =2
(5.9)
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18
j 2ν + 2 K 2 f 2H ν ≤ 2(1−4α) exp − 2 + 1 n =2 ≤
K 2 b2 f 2H ν K 2 b2 f 2 ν exp(−(ν + 1) ln j) = 2(1−4α) H , 2(1−4α) n n j ν+1
for some positive constant b, independent of j. Thus
[(n−2)/2]
Kbf H ν 2j j! ≤ (2j + 1)! n1−4α
I2j+1,n
j=2
[(n−2)/2]
1 j
j=2
≤
ν+1 2
Kf H ν , n1−4α
(5.10)
where K is a generic constant independent of n. By (5.8)–(5.10), n−1
Im,n |Hm (x)|G(x)/m! ≤
m=0
Kf H ν , for all x ∈ R. n1−4α
(5.11)
H Since K f n1−4α converges to 0 as n → ∞, the sequence of partial sums on the left converges uniformly on R. On the other hand, ν
Jm,n ≤ J1,m,n + J2,m,n ,
(5.12)
where
m −u2 /2
|u| e
J1,m,n :=
|f(u)|du,
J2,m,n :=
|u|>nα
|u| M n (u) f(u) du. m
|u|>nα
By the Cauchy–Schwarz inequality, ⎧ ⎪ ⎨
J1,m,n ≤
|u|2(m−ν) e−u du 2
⎪ ⎩
|u|>nα
⎧ ∞ ⎨ =
⎩
tm−ν−1/2 e−t dt
n2α
where f H ν α := n
|u|>nα
|f(u)|2 |u|2ν du
n−1
⎫1/2 ⎧ ⎪ ⎬ ⎪ ⎨ ⎪ ⎭
⎫1/2 ⎬ ⎭
|f(u)|2 |u|2ν du
⎪ ⎩
|u|>nα
⎫1/2 ⎪ ⎬ ⎪ ⎭
f H ν α ≤ Γ(m − ν + 1/2)1/2 f H ν α , n
n
1/2 . As in the proof of (5.11),
J1,m,n |Hm (x)|G(x)/m! ≤ Kf Hnν α , for all x ∈ R.
(5.13)
m=0
By Lemma 3.2,
J2,m,n ≤
⎧ ⎪ ⎨ ⎪ ⎩
|u|>nα mα
≤
n
e
⎫1/2 2 ⎪ ⎬ (m−ν) n (u) du u f Hnν α M ⎪ ⎭
f Hnν α 1 2α 2n
⎧ ⎪ ⎨ ⎪ ⎩
|u|>nα
⎫1/2 ⎪ ⎬
1 du |u|2ν ⎪ ⎭
≤
Knα(m−ν+1/2) f Hnν α 1
e2n
2α
,
(5.14)
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19
for all sufficiently large n, where K is a generic constant that is independent of m, n. It follows from (2.8) and (5.14) that n−1
J2,m,n |Hm (x)|G(x)/m! ≤
m=0
n−1
Kf Hnν α 1
nα(ν−1/2) e 2 n
nαm (m−1)/2 2 Γ m! m=0
2α
m+1 2
,
and since
2j j!, 2(m−1)/2 Γ ((m + 1)/2) = (2j)! π
m = 2j + 1,
2,
2j j!
m = 2j,
this gives n−1
J2,m,n |Hm (x)|G(x)/m! ≤
m=0
≤ ≤
⎛
Kf
ν Hn α α(ν−1/2) 12 n2α
n
e
1
j=0
[(n−1)/2]
2α
Kf Hnν α nα(ν−1/2) e
⎝ ⎛
Kf Hnν α nα(ν−1/2) e 2 n
[(n−1)/2]
1 2α 2n
⎝
j=0
1
e2n
2α
2jα
n + 2j j!
[(n−2)/2]
1
2α
n
j
2 j! ⎠ (2j + 1)!
j=0
n2jα + nα 2j j!
+ nα e 2 n
⎞ (2j+1)α
[(n−2)/2]
j=0
⎞ n2jα ⎠ 2j j!
.
(5.15)
Hence, by (5.12), (5.13) and (5.15), for all x ∈ R, n−1
Jm,n |Hm (x)|G(x)/m! ≤ 3Kf Hnν α .
(5.16)
m=0
Combining (5.11) and (5.16) gives n−1
(Im,n + Jm,n )|Hm (x)|G(x)/m! ≤
m=0
Kf H ν + 3Kf Hnν α , n1−4α
for all x ∈ R. Since α < 1/4, the sequences on the right converge to 0 as n → ∞, and so the sequence of partial sums on the left converges uniformly on R. By (5.6), An f (x) → 0 locally uniformly as n → ∞. Finally we show that Bn f (x) → 0 locally uniformly on R. By (4.3), n−1 K n−1 m/2 n,m (x) Hm (x) − B m (m) −x2 n(m) , f e e (−1) Mn , f , ≤ (−1)m M m! n m=0 2m m=0 n , f = where K is a generic constant that is independent of m, n. Since (−1)m M (m)
∞ −∞
n (u)f(u)du, (iu)m M
∞ m m (m) (−1) Mn , f ≤ u M n (u) f (u) du −∞ α n m f(u) du + u = (u) M n
−nα
|u|>nα
m n (u) f(u) du, u M
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20
where 0 < α < 1/2. So n−1 n,m (x) 2 (x) − B H m (m) , f e−x ≤ Tn,1 + Tn,2 , (−1)m M n m!
(5.17)
m=0
where
Tn,1
α n n−1 m K e m/2 n (u) f(u) du, u M := n m=0 2m
(5.18)
−nα
Tn,2
n−1 K e m/2 := n m=0 2m
m n (u) f(u) du. u M
(5.19)
|u|>nα
By (3.1) and as in the derivation of (5.7), α α n n m 2 n (u) f(u) du ≤ u M |u|m e−u /2 f(u) du ≤ Kf H ν Γ(m − ν + 1/2)1/2 ,
−nα
−nα
and so by (5.18), Tn,1 ≤
n−1 e m/2 Kf H ν Γ(m − ν + 1/2)1/2 . n 2m m=0
The series on the right converges as n → ∞ by the ratio test, and so Tn,1 ≤
Kf H ν . n
(5.20)
Also as in the proof of (5.14) and (5.15), (5.19) leads to Tn,2 ≤
1 2α e m/2 K e− 2 n f Hnν α n−1 nmα 1+α(ν−1/2) 2m n m=0
K e− 2 n f Hnν α ≤ n1+α(ν−1/2) 1
2α
(1/2 n−1
n−1 ( e m 1/2 n2mα m! m! 2m m=0 m=0
K e− 2 n f Hnν α 1 n2α ≤ e2 n1+α(ν−1/2) 1
since
n−1 m=0
m!
e m 2m
2α
n−1
e m m! 2m m=0
(1/2 ≤
Kf Hnν α , n1+α(ν−1/2)
(5.21)
converges as n → ∞. Combining (5.17), (5.20) and (5.21) gives
n−1 Kf ν n,m (x) Kf Hnν α 2 H (x) − B m H (m) , f e−x ≤ + 1+α(ν−1/2) (−1)m M , for all x ∈ R. n m! n n m=0
(5.22)
It follows that the sequence of partial sums on the left of (5.22) converges to 0 uniformly on R as n → ∞. Hence, Bn f (x) in (5.4) converges locally uniformly on R. 2
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Proof of Theorem 1.1. For x, t ∈ R, s > 0, let fs,t (x) := f (t − x/s). Then fs,t (u) = sf(−us)e−iust , u ∈ R, and fs,t ∈ H ν (R) if f ∈ H ν (R). As in (5.1)–(5.4), fs,t (x) =
n−1
(m) , fs,t (−1)m M n
m=0
n,m (x) B + Rn fs,t (x), m!
where Rn fs,t (x) =
∞
(−1)m G(m) , fs,t Hm (x)/m! −
m=0
n−1
n,m (x)/m! (m) , fs,t B (−1)m M n
m=0
= An fs,t (x) + Bn fs,t (x) +
∞
(−1)m G(m) , fs,t Hm (x)/m!.
(5.23)
m=n
By (2.12), ∞ ∞ 2 1 m (m) (−1) G , fs,t Hm (x)/m! ≤ Cfs,t H ν ex /2 . (1+ν)/2 m m=n m=n
(5.24)
As in the proof of Theorem 5.1, for α < 1/4, 2
|An fs,t (x)| ≤ K ex
) /2
fs,t H ν + 3fs,t Hnν α n1−4α
* ,
(5.25)
.
(5.26)
and for α < 1/2, |Bn fs,t (x)| ≤ K e
x2
)
fs,t Hnν α fs,t H ν + 1+α(ν−1/2) n n
*
Since
fs,t H ν ≤
1 f H ν , sν−1/2 1/2 s f H ν ,
0 < s < 1, s ≥ 1,
and fs,t Hnν α =
1 sν−1/2
ν f Hsn α,
it follows from (5.23)–(5.26) that Rn fs,t (x) converges to 0 uniformly for t ∈ R and locally uniformly for x ∈ R and s > 0. 2 6. Approximation of derivatives of B-splines by spline framelets In this section we show that the spline framelets constructed by Ron and Shen [3], when suitably standardized, approximate the derivatives of the standardized B-splines and converge to the derivatives of the 1 Gaussian function as the orders of the spline functions tend to infinity with a convergence rate of O( n1−/p ), p 0 < < 1, in the L -norm, 1 ≤ p ≤ ∞. Bernoulli–Schoenberg series expansion formulas of the form (1.5) and (5.1) are then derived with the standardized spline framelets as mother wavelets instead of the derivatives of the B-splines in the wavelet transforms.
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22
n is The Fourier transform of the standardized B-splines M n (u) = M
n sin(u 3/n) , u 3/n
u ∈ R.
For integers n ≥ 1, m = 0, 1, . . . , n, we define ψ n,m (u) :=
4n 3
= (iu)
m/2 im
m
sin(u
m 3/4n)
n−m n (u/2) cos(u 3/4n) M
n−m 2m sin(u 3/n) sin(u 3/4n) , u 3/n u 3/4n
u ∈ R.
(6.1)
For even n, ψ n,m are, up to constant multiples and a constant dilation, the spline framelets of Ron and Shen [3]. They are also, up to constant multiples and a constant dilation, the mother wavelets for the semidiscrete wavelet transforms arising from the B-spline scale-space considered in [2]. We shall call ψ n,m the n(m) of the standardized standardized Ron–Shen framelets, and show that they approximate the derivatives M B-splines and also converge to the derivatives of the Gaussian function as n → ∞. Theorem 6.1. For any nonnegative integer m,
1 (m) as n → ∞. sup ψ n,m (u) − Mn (u) = O n u∈R
(6.2)
Proof. The Fourier transforms of the derivatives of the standardized B-spline can be expressed as
n sin(u 3/n) u 3/n n−m m m sin u 3/n sin(u 3/4n) m = (iu) . cos(u 3/4n) u 3/n u 3/4n
(m) m M n (u) = (iu)
By (6.1) and (6.3), n−m m sin(u 3/4n) sin(u 3/n) n(m) (u) = um ψ n,m (u) − M Fm,n (u) u 3/n u 3/4n n−m m sin(u 3/n) Fm,n (u), ≤ u u 3/n where sin(u3/4n) m m Fm,n (u) := − cos(u 3/4n) u 3/4n sin(u3/4n) ≤ m − cos(u 3/4n) u 3/4n ∞ 2j Kmu2 2j = m (−1)j−1 , u 3/4n ≤ (2j + 1)! n j=1
(6.3)
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for all positive integers n ≥ m and u ∈ R, where K is a constant independent of m, n and u. Hence n−m 3/n) (m) , u ∈ R. Km um+2 sin(u ψ n,m (u) − M n (u) ≤ n u 3/n For |u| ≤
√
(6.4)
n, the proof of (3.1) in Lemma 3.1 gives n−m m+2 sin(u 3/n) u ≤ |u|m+2 e−(n−m)u2 /2n . u 3/n +
The maximum of the expression on the right is attained at u = n > 2(m + 1), and so
n(m+2) n−m ,
(6.5)
which is less than
n−m
(m+2)/2 m+2 sin(u 3/n) m+2 u ≤ e−(m+2)/2 . 1 − m/n u 3/n
√
n for
(6.6)
For |u| ≥ nα , 0 < α < 1/2, similar arguments as in the proof of Lemma 3.2 show that if n > 2(m + 1), n−m m+2 sin(u 3/n) u ≤ nα(m+2) e−n2α /2 em/2n1−2α ≤ Knα(m+2) e−n2α /2 u 3/n
(6.7)
for all sufficiently large n, where K is a generic constant independent of n and u. The estimate (6.2) now follows from (6.4), (6.6) and (6.7). 2 Corollary 6.2. For any nonnegative integer m,
1 m −u2 /2 =O as n → ∞. sup ψ m,n (u) − (iu) e n u∈R Proof. The result follows from Theorems 3.3 and 6.1. 2 Theorem 6.3. For any integer m ≥ 0, (m) ψn,m − M n
∞
=O
1 as n → ∞. n
(6.8)
Proof. By taking inverse Fourier transforms, for any positive integer n ≥ m and x ∈ R, (m) (x) = 1 ψ n,m (x) − M n 2π
∞ e
iux
(m) ψ m,n (u) − Mn (u) du
−∞
and so ∞ 1 (m) (m) ψ m,n (u) − Mn (u) du ψn,m (x) − Mn (x) ≤ π 0 n−m ∞ 3/n) K um+2 sin(u du, ≤ πn u 3/n 0
(6.9)
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by (6.4). For a positive α < 1/2, a similar argument as in the proof of Theorem 3.4 using (6.5) gives α α n−m n n m+2 sin(u 3/n) u du ≤ um+2 e−(n−m)u2 /2n du u 3/n
0
0
≤ 2m+2 Γ((m + 3)/2), for n > 2m,
(6.10)
and n−m ∞ m+2 sin(u 3/n) u du ≤ C n(m+3)α e−n2α /2 , u 3/n α
(6.11)
n
where C is a constant independent of n. It follows from (6.9)–(6.11) that (m) (x) ≤ K 2m+2 Γ((m + 3)/2) + CK n(m+3)α e−n2α /2 , ψn,m (x) − M n πn πn which gives (6.8). 2 Corollary 6.4. For any integer m ≥ 0, ψn,m − G(m)
∞
=O
1 as n → ∞. n
(6.12)
Proof. The estimate (6.12) follows from Theorems 3.4 and 6.3. 2 Remark 3. The problem of approximating the spline framelets of Ron and Shen by derivatives of the Gaussian function is also considered recently by Shen and Xu in [5]. However, their results are different from ours. They consider the nonstandardized Gaussian function ,
n 12 12t2 , Gn, (t) := exp − 2n −
π(2n − )42
t ∈ R,
d and show that for large n and = 1, 2, . . . , n, the derivatives dt Gn, (t − jn /2), where jn = 0 or 1 depending on n is even or odd respectively, approximate the original nonstandardized spline generators ψn, uniformly. An error estimate is also given.
We now give the rate of convergence in the Lp -norm. Theorem 6.5. Take any integer m ≥ 0 and 1 ≤ p < ∞. Then for any 0 < < 1, (m) = O ψn,m − M n p
1
n1−/p
as n → ∞.
(6.13)
Proof. By Theorem 6.3, for any positive < 1 and for all positive integers n ≥ m, n n(m) (x) dx ≤ 2n n(m) ψn,m (x) − M ψ n,m − M
−n
for some constant C > 0. Then for all positive integers n ≥ m,
∞
≤
C , n1−
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n n n(m) (x)dx ≤ C . ψ n,m (x)dx − M n1− 0
25
(6.14)
0
Now for even m, ∞
(m) (x)dx M n
∞ =
0
ψ n,m (x)dx =
0
1 2,
m = 0,
0,
m > 0.
For odd m, defining gn,m−1 by gn,m−1 (u) := (iu)
m−1
n−m 2m sin(u 3/n) sin(u 3/4n) , u 3/n u 3/4n
we have, by (6.1), its derivative gn,m−1 = ψ n,m . Hence
∞
ψ n,m (x)dx −
0
∞
(m−1) (0). (m) (x)dx = −gn,m−1 (0) + M M n n
0
The proof of Theorem 6.3 gives (m−1) gn,m−1 − M n
∞
=O
1 , n
and so for odd m, ∞
ψ n,m (x)dx −
0
∞
(m) (x)dx = O M n
1 . n
0
Therefore, for any integer m ≥ 0, ∞ ∞ (m) (x)dx ≤ C . ψ n,m (x)dx − M n n 0
0
It follows from (6.14) and (6.15) that ∞
∞ 1 (m) (x)dx = O ψ n,m (x)dx − M . n n1− n
n
Then similar arguments as in the proof of Theorem 3.5 give
1 (m) , = O ψn,m − M n n1− 1 and hence (6.13). 2
(6.15)
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Corollary 6.6. Take any integer m ≥ 0 and 1 ≤ p < ∞. Then for any 0 < < 1, (m) ψn,m − G = O p
1 n1−/p
as n → ∞.
Proof. The result follows from Theorems 3.5 and 6.5. 2 n(m) of the standardized Since the standardized spline framelets ψ n,m approximate the derivatives M B-splines, it is of interest to investigate to what extent the latter can be replaced by the former in (1.5) and (5.1) of Theorems 1.1 and 5.1 respectively. The next theorem gives a result of this form. Theorem 6.7. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then
f (x) =
√ 0≤m< n
(−1)m ψ n,m , f
n,m (x) B n f (x), +R m!
n f (x) → 0 as n → ∞ locally uniformly for x ∈ R. where R Proof. The proof uses Theorem 5.1 and similar techniques as in its proof as well as the results on the approximation of the derivatives of the standardized B-splines by the standardized spline framelets. If √ f ∈ H ν (R), ν ≥ 3/2, and is continuous, by the proof of Theorem 5.1 (with the sum over 0 ≤ m < n replacing the sum over 0 ≤ m ≤ n − 1), f (x) =
√ 0≤m< n
(m) , f (−1)m M n
n,m (x) B + Rn f (x), m!
n , this gives where Rn f (x) → 0 as n → ∞ locally uniformly for x ∈ R. Recalling that ψ n,0 = M f (x) =
(−1)m ψ n,m , f
√ 0≤m< n
−
n,m (x) B m!
n(m) , f Bn,m (x) + Rn f (x). (−1)m ψ n,m − M m!
√ 1≤m< n
Let Sn (x) :=
√ 1≤m< n
(m) , f Bn,m (x) , (−1)m+1 ψ n,m − M n m!
n f (x) := Sn (x) + Rn f (x). R
We want to show that Sn (x) → 0 locally uniformly for x ∈ R, as n → ∞. Applying Parseval’s identity,
∞ n,m (x)| 1 |B (m) |Sn (x)| ≤ ψ n,m (u) − Mn (u) f (u)du 2π m! √ 1≤m< n
−∞
≤
1 2π
√ 1≤m< n
⎧ nα ⎨ (m) (u) − M ψ f (u) (u) du n n,m ⎩ −nα
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+ |u|>nα
=
√ 1≤m< n
27
⎫ ⎪ ⎬ (m) f(u) du |Bn,m (x)| ψ n,m (u) − M (u) n ⎪ m! ⎭ (Im,n,1 + Im,n,2 )
n,m (x)| |B , m!
(6.16)
where
Im,n,1
1 := 2π
Im,n,2 :=
and 0 < α < 1/4. For 1 ≤ m <
√
1 2π
α n (m) f(u) du, ψ n,m (u) − M (u) n
−nα
(m) ψ n,m (u) − M f (u) (u) du, n
|u|>nα
n, by Proposition 4.2, B e m/2 2 n,m (x) − Hm (x) K ex ≤ m! n 2m
where K is a generic constant independent of m, n, and so this together with (2.9) give n,m (x)| K e m/2 x2 |Hm (x)| |B ≤ e + m! n 2m m! m/2 2 e K K e m/2 x2 /2 ≤ ex + √ e n 2m m m K e m/2 x2 ≤√ e . m m
(6.17)
Now,
Im,n,1
1 = 2π
α n (m) f(u) du ψ n,m (u) − M (u) n
−nα
⎫ ⎧ nα 2 ⎬1/2 f H ν ⎨ (m) ≤ √ (1 + |u|2 )−ν ψ n,m (u) − M . n (u) du ⎭ 2π ⎩ α −n
By (6.4) and using the same method as in the proof of (6.10),
Im,n,1 ≤
⎧ nα ⎨
Kmf H ν ⎩ n
|u|2m+4−2ν
−nα
≤
Kmf n
Hν
Kmf H ν ≤ n
sin(u 3/n) u 3/n
⎫1/2 ⎬
2(n−m) du
⎧ nα ⎫1/2 ⎨ ⎬ (n−m) 2 u2m+4−2ν e− n u du ⎩ ⎭
0
n n−m
(2m+5−2ν)/4 Γ(m + 5/2 − ν)1/2 .
⎭
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For m <
√
n, the second factor in the last expression is bounded, and so
Kmf H ν Γ(m + 5/2 − ν)1/2 . n √ Also by (6.4) and (6.7), for |u| > nα , m < n and α < 1/4, Im,n,1 ≤
(6.18)
Km α(m+2) −n2α /2 m/2n1−2α (m) ≤ ψ n,m (u) − M n (u) e e n n ≤
K n(1−4α)/2
nmα e−n
2α
/2 1/2n(1−4α)/2
e
K
≤
n(1−4α)/2
nmα e−n
2α
/2
,
and so ⎧ ⎪ ⎨
f H ν Im,n,2 ≤ √ 2π ⎪ ⎩
|u|>nα
≤
⎫1/2 2 ⎪ ⎬ (m) |u|−2ν ψ n,m (u) − M n (u) du ⎪ ⎭ ⎧ ⎪ ⎨
Kf H ν mα −n2α /2 n e ⎪ n(1−4α)/2 ⎩
|u|−2ν du
|u|>nα
≤
⎫1/2 ⎪ ⎬ ⎪ ⎭
Kf H ν α(m−ν+1/2) −n2α /2 n e . n(1−4α)/2
(6.19)
It follows from (6.16)–(6.19) that |Sn (x)| ≤
Kf H ν n +
√ 1≤m< n
Kf H ν n(1−4α)/2
1 e m/2 x2 mΓ(m + 5/2 − ν)1/2 √ e m m
√ 1≤m< n
nα(m−ν+1/2) e−n
2α
/2
1 e m/2 x2 √ e =: Sn,1 (x) + Sn,2 (x). m m
Noting that for ν ≥ 3/2, Γ(m + 5/2 − ν) ≤ Γ(m + 1), and applying the Stirling formula,
m m/2 e m/2 2 ex e m
Sn,1 (x) ≤
Kf H ν n
=
Kf H ν n
≤
Kf H ν x2 e → 0 locally uniformly for x ∈ R. n1/16
√ 1≤m< n
√ 1≤m< n
m3/4
Kf H ν m15/8 x2 e ≤ m9/8 n1/16
√ 1≤m< n
1 x2 e m9/8
Applying exactly the arguments of the proof of (5.15) for the second sum, Kf H ν 1 + nα x2 e n(1−4α)/2 nα(ν−1/2) Kf H ν 2 ≤ (1−4α)/2 ex , n
|Sn,2 (x)| ≤
n f (x) converge locally uniformly to 0 as n → ∞. 2 since ν ≥ 3/2. Therefore Sn,2 (x) and hence Sn (x) and R
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We conclude with the following theorem, which can be derived from the proof of Theorem 6.7 in the same way as Theorem 1.1 from the proof of Theorem 5.1. Theorem 6.8. If f ∈ H ν (R) for some ν ≥ 3/2 and is continuous, then for x, t ∈ R, s > 0, f (x) = SM n f (s, t) +
√ 1≤m< n
n,m (s(t − x)) (−1)m B n f (x, s, t), Wψ n,m f (s, t) + R m!
n f (x, s, t) → 0 as n → ∞ locally uniformly for x ∈ R and s > 0 and uniformly for t ∈ R, and where R SM n,m f (s, t) are continuous wavelet transforms of f n f (s, t) is the B-spline scale-space transform of f , Wψ with the standardized spline framelets as mother wavelets. References [1] R. Brinks, On the convergence of derivatives of B-splines to derivatives of the Gaussian function, Comput. Appl. Math. 27 (2008) 79–92. [2] S.S. Goh, T.N.T. Goodman, S.L. Lee, Singular integrals, scale-space and wavelet transforms, J. Approx. Theory 176 (2013) 68–93. [3] A. Ron, Z. Shen, Affine systems in L2 (Rd ): the analysis of the analysis operator, J. Funct. Anal. 148 (1997) 408–447. [4] I.J. Schoenberg, Contributions to the problem of approximation of equidistant data by analytic functions, Quart. Appl. Math. 4 (1946) 45–99, 112–141. [5] Z. Shen, Z. Xu, On B-spline framelets derived from the unitary extension principle, SIAM J. Math. Anal. 45 (2013) 127–151. [6] G. Szegö, Orthogonal Polynomials, 4th edition, AMS Colloquium Publications, vol. XXIII, American Mathematical Society, 1975. [7] Y. Xu, R. Wang, Asymptotic properties of B-splines, Eulerian numbers and cube slicing, J. Comput. Appl. Math. 236 (2011) 988–995. [8] J.M. Whittaker, On Lidstone’s series and two-point expansions of analytic functions, Proc. Lond. Math. Soc. 36 (1934) 451–469.