Pointwise convergence for expansions in spherical monogenics

Acta Mathematica Scientia 2009,29B(5):1241–1250 http://actams.wipm.ac.cn

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS∗ Fei Minggang (


)1

)2

Qian Tao (

1.School of Applied Mathematics, University of Electronic Science and Technology of China, Changdu 610054, China 2.Department of Mathematics, Faculty of Science and Technology, University of Macau, Macao, China E-mail: [email protected]; [email protected]

Abstract We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter’s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson’s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson’s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres. Key words spherical monogenics; pointwise convergence of Fourier-Laplace series; generalized Cauchy-Riemann operator; unit sphere; generalization of Fueter’s theorem 2000 MR Subject Classification

1

42B05; 30G35

Introduction

We start with reviewing the basic notations and results. Let f ∈ L1 ([−π, π]), then the Fourier coefficients ck are all well defined by  π 1 ck = f (t)e−ikt dt, k ∈ Z, (1.1) 2π −π where Z denotes the set of all integers. By sN (f )(x), we denote the partial sum sN (f )(x) =



ck eikx ,

x ∈ [−π, π], N ∈ N0 ,

(1.2)

|k|≤N ∗ Received

04S/QT/FST

August 23, 2007. Sponsored by Research Grant of the University of Macau No. RG024/03-

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of the Fourier series of f , where N0 denotes the set of all natural numbers. Then we have,  1 π sN (f )(x) = f (t)DN (x − t)dt, (1.3) π −π where ⎧ sin(N + 12 )x ⎪ ⎪ ⎨ for x ∈ [−π, π]\{0}, 2 sin x2 DN (x) = ⎪ ⎪ ⎩N + 1 for x = 0 2 is called the N -th Dirichlet kernel. Since L2 ([−π, π]) ⊂ L1 ([−π, π]), the Fourier coefficients of L2 -functions are well defined. There holds the following Carleson’s theorem. Theorem 1.1 [1] If f ∈ L2 ([−π, π]), then sN (f )(x) → f (x),

a.e. x ∈ [−π, π], as N → +∞.

L. Carleson proved this theorem in 1966. The next year, R.A. Hunt [7] further extended this result to f ∈ Lp ([−π, π]), 1 < p < ∞. One naturally asks what is the analogous result for the unit sphere S n in the (n + 1)dimensional Euclidean space Rn1 , where Rn1 = {x = x0 + x | x0 ∈ R, x ∈ Rn }? For any square integrable function f defined on S n , denoted by f ∈ L2 (S n ), there is an associated Fourier-Laplace series: f∼

∞ 

fk ,

(1.4)

k=0

where fk is the homogeneous spherical harmonics of degree k. There were comprehensive studies on convergence and summability of Fourier-Laplace series on the unit sphere of higher dimensional Euclidean spaces (see [8], [17], [20]). However, except for the two-dimensional case, pointwise convergence, being the initial motivation of harmonic analysis, may be said to be very little known. The case n = 1 seems to be the only well studied case ([1], [21]). Dirichlet ([2]) made the first detailed study for the case n = 2, on the so-called Laplace series. Koschmieder ([22]) studied the case n = 3. Roetman ([17]) considered the general cases, and, under certain conditions, reduced the convergence problem for n = 2k + 1 to n = 1; and n = 2k + 2 to n = 2. Among others, Meaney ([10]) and Yu ([23]) addressed some related topics, including the Lp cases. The theory of Fourier-Laplace series is not facilitated with a complex structure like what is in the complex plane. All the known studies on Fourier-Laplace series are heavily dependent on the properties of spherical harmonics, especially Legendre polynomials. In this study, we provide a new approach based on Clifford algebras. Based on the induced complex structure we obtain integral expressions of partial sums in terms of Dirichlet kernels where no knowledge of special functions is involved. Our approach is based on generalizations of Fueter’s theorem

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(or inducing theorem, see below) on inducing monogenic and harmonic functions from those for one complex variable. In particular, the Dirichlet kernels (see Section 3) are induced from those for one complex variable by using inducing theorem. The convergence results obtained here are stronger than those obtained in [17] with simpler proofs, while [17] heavily relies on the special functions machinery. The applicability of the method adopted in this article is restricted to the even dimensional cases. That is because, the Fueter’s theorem in those cases reduces to pointwise differentiation, and further reduces to the problems in complex plane. Our proofs are based on this pointwise differentiation approach. On the other hand, for the odd dimensional cases, Fueter’s theorem reduces to computation on Fourier multiplier operators, with connections to complex plane; or alternatively to pointwise differentiation but based on the three-dimensional Euclidean space. The odd dimensional cases would require different ideas ([11], [13], [15]). This article is a further development of [5] and [9]. In [5], the authors obtained the detailed expression of the Dirichlet kernels restricted on the unit sphere. Then, Riemann-Lebesgue theorem, localization principle, and a Dini’s type pointwise convergence theorem are proved. In this article, based on the classical Carleson’s theorem, we further obtain a weaker condition in terms of certain Sobolev spaces that guarantees the pointwise convergence of the FourierLaplace series (1.4) (see Section 4 for details). The result is a generalization of Carleson’s theorem to the higher dimensional Euclidean spaces. We include here some propositions and results from [5] without proofs.

2

Preliminaries

We will be working with Rn1 , the real-linear span of e0 , e1 , · · · , en , where e0 is identical with 1 and ei ej + ej ei = −2δij . Rn1 is embedded into the Clifford alegbra R(n) generated by e1 , · · · , en . A typical element in Rn1 is denoted x = x0 + x, where x0 ∈ R and x = x1 e1 + · · · + xn en ∈ Rn . We will study Rn1 -variable and Clifford-valued functions and the concepts of left- and right-monogenic functions are introduced via the generalized Cauchy∂ ∂ Riemann operator D = ∂x e0 + ∂x e1 + · · · + ∂x∂ n en : a function f with continuous first order 0 1 derivatives is said to be left-monogenic or right-monogenic if Df = 0 or f D = 0 in its domain, respectively. A function is said to be monogenic, if it is both left- and right-monogenic. The n+1

¯ 2π 2 Cauchy kernel is denoted by E(x) = ω1n |x|xn+1 , x ∈ Rn1 \ {0}, where ωn = Γ( is the surface n+1 2 ) area of the n-dimensional unit sphere S n in Rn1 . The Cauchy kernel is both left- and rightmonogenic. Let f 0 be a complex-valued function defined in an open set O in the upper-half complex plane. Write f 0 (z) = u(t, s) + iv(t, s), where u and v are real-valued.  Denote, for x ∈ O, x v(x0 , |x|), f0 (x) = u(x0 , |x|) + |x|

 = {x ∈ Rn |(x0 , |x|) ∈ O}. where O 1  0  is the induced set form O. The f is said to be the induced function form f 0 , and O concept of intrinsic sets and functions naturally fit into our theory. In the complex plane C, a set is said to be intrinsic if it is open and symmetric with respect to the real axis; and a

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function f 0 an intrinsic function if it is defined in an intrinsic set and satisfies f 0 (z) = f 0 (¯ z) within its domain (see [11]). In the notation f 0 = u + iv, the above condition is equivalent to require that u is even and v is odd in their second argument. In particular, v(x0 , 0) = 0, i.e., f 0 is real-valued if it is restricted to the real line in its domain. Denote by τ the mapping n−1 τ (f 0 ) = κ−1 Δ 2 f0 , ¯ D ¯ = ∂ e0 − ∂ e1 − · · ·− ∂ en and κn = (2i)n−1 Γ2 ( n+1 ) are the normalizing where Δ = DD, ∂x0

∂x1

∂xn

2

constant that makes τ ((·)−1 ) = E (see [14]). n−1 The operator Δ 2 is defined via the Fourier multiplier transformation on tempered distributions M : S  → S  induced by the multiplier m(ξ) = (2πi|ξ|)n−1 : M f = R(mFf ), 

where F f (ξ) =

e2πix,ξ f (x)dx, Rn 1



and Rh(x) =

e−2πiξ,x h(ξ)dξ. Rn 1

It is noted that both the Fourier transformation F and its inverse R are defined on tempered distributions via pairing with rapidly decreasing functions. n−1 If n is an odd integer, then Δ 2 reduces to a point-wise differential operator that was first studied by M. Sce who extended Fueter’s result to Rn1 for odd n ∈ Z+ ([18]). The corresponding result for n even was obtained and discussed by [13] and [14]. We have the following ([6], [18], [12–14]). Proposition 2.1 Let f 0 (z) = u(s, t) + iv(s, t) be an intrinsic function defined on an  intrinsic set O ⊂ C. Then, the function τ (f 0 ) is monogenic in O. If we consider f 0 to be of the form z k , k ∈ Z, then we define the monomial functions by P (−k) = τ ((·)−k ),

P (k−1) = I(P (−k) ),

k ∈ Z+ ,

where Z+ is the set of positive integers and I is the Kelvin inversion defined by I(f )(x) = E(x)f (x−1 ). When n is odd, the striking facts ([14]) are τ ((·)(k) ) = P (k) , k = −1, −2, · · · ;

(2.1)

τ ((·)(k+n−1) ) = P (k) , k = 0, 1, 2, · · · .

(2.2)

We note that, for any k, P (k) (y −1 x)E(y) is monogenic both in x and y ([14]). Since E(y)n(y) = 1 on the sphere, by Cauchy’s theorem, we have  P (k) (y −1 x)dσ(y) = 0, |x| = 1, k = 0; (2.3) Sn

and, since P

(0)

= I(P

(−1)

) = I(E) = 1, we have  1 P (0) (y −1 x)dσ(y) = 1, ωn S n

|x| = 1.

(2.4)

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Expansions of the Fourier-Laplace Series and Dirichlet Kernels

In the frame of Clifford algebras, we are enable to decompose spherical harmonics into sums of spherical monogenics ([4]). For any k-spherical harmonic fk , as appeared in the FourierLaplace series of f ∈ L2 (S n ) in (1.4), one has the unique decomposition f k = g k + hk ,

(3.1)

where gk is the restriction on the sphere of a left-monogenic function of homogeneity k, and hk the same restriction of a left-monogenic function of homogeneity −k + 1 − n ([3], [4]). Results in [14] imply that  1 gk (x) = P (k) (y −1 x)f (y)dσ(y), ωn S n and

1 hk (x) = ωn



P (−k) (y −1 x)f (y)dσ(y).

Sn

Note that the expressions P (k) (y −1 x) make sense as the domain of P (k) may be extended to products of vectors ([14]). The partial sum corresponding to (1.4), denoted by SN f (x), may be expressed as  1 (n+1) −1 SN f (x) = D (y x)f (y)dσ(y), (3.2) ωn S n N where (n+1)

DN

(x) =



P (k) (x)

(3.3)

|k|≤N

is called the N -th Dirichlet kernel in Rn1 . When n is an odd number, invoking (2.1) and (2.2), (n+1)

DN

(x) = τ

+n−1  N (·)(k) (x).

(3.4)

k=−N

In this article, we only consider n being odd numbers and therefore (n − 1)/2 being positive (n+1) integers. The computation of DN (x) is based on the inducing theorem obtained in [15], as cited in the following proposition. Proposition 3.1 Let h(t, s) be harmonic for t and s in a region O in which s > 0, t > 0. Let n ∈ Z+ being odd and x = x0 + x ∈ Rn1 . Define H(x) = Δ(n−1)/2 h(x0 , |x|), where the Laplacian Δ is in the n + 1 variables, then 1 H(x) = (n − 1)!!( ∂s )(n−1)/2 h(t, s)|t=x0 ,s=|x| , s

(3.5)

and H is harmonic in x0 , x1 , · · · , xn in the corresponding region in Rn1 . Without loss of generality, we assume that functions to be expanded in (1.4) are scalarvalued. Indeed, a Clifford-valued function may be separated into 2n parts of each which is scalar-valued.

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Assume that f ∈ L2 (S n ) and f is associated with an expansion (1.4). From the decomposition (3.1) and the integral formulas for gk and hk , we have  1 (P (k) + P (−k) )(y −1 x)f (y)dσ(y). fk (x) = ωn S n Since f is scalar-valued, the scalar part of P (k) + P (−k) will produce fk , and the nonscalar parts of P (k) and P (−k) will have to be canceled out. This concludes that only the scalar parts of the (n+1) (n+1) Dirichlet kernels are concerned. We denote the scalar part of DN by DN . Denote 0 fN (z) =

N +n−1

z k = z −N + · · · + z −1 + 1 + z + · · · + z N +n−1 ,

k=−N 0 (z) = UN (t, s) + iVN (t, s), where z = t + is, t, s ∈ R. Thus, both UN and VN are and fN harmonic functions in t and s. Since we only consider n being odd numbers, we write n = 2l + 1, l ∈ Z+ for the rest of (n+1) (2l+2) (2l+2) the article. Then DN = DN . From [5], we have the detailed expression of DN . n Proposition 3.2 For any x ∈ S , i.e., x = x0 + x1 + · · · + xn , and |x| = 1. Then (2l+2)

DN

(x) = κ−1 n (2l)!!

j l   j=1 i=1

+κ−1 n (2l)!!

(j)

Clj (sin θ)j−2l Qi (sin θ, cos θ)(

j l   j=1 i=1

(j)

(j)

∂ i ) UN ∂θ

Clj (sin θ)j−2l Ri (sin θ, cos θ)(

∂ i ) VN , ∂θ

(j)

(3.6)

where θ = arccos x0 , Qi and Ri are homogeneous polynomials of degree j in two variables, Clj = (−1)l−j (2l−j−1)!(2l−2j−1)!! , 1 ≤ j ≤ l − 1; Cll = 1. (j−1)!(2l−2j)! Now, we work out UN and VN in polar coordinates. We have 0 fN (z) =

N +n−1 k=−N

zk =

z N +n 1 − z N (1 − z) 1−z

cos N θ − r cos(N + 1)θ − r2N +n cos(N + n)θ + r2N +n+1 cos(N + n − 1)θ = rN (r2 − 2r cos θ + 1) − sin N θ + r sin(N + 1)θ − r2N +n sin(N + n)θ + r2N +n+1 sin(N + n − 1)θ +i rN (r2 − 2r cos θ + 1) = UN (r cos θ, r sin θ) + iVN (r cos θ, r sin θ). Restricting to the unit sphere, we have UN (cos θ, sin θ) = =

cos N θ − cos(N + 1)θ − cos(N + n)θ + cos(N + n − 1)θ 2(1 − cos θ) sin 2N2+1 θ + sin 2N +2n−1 θ 2 2 sin

θ 2

=

sin(N + n2 )θ cos n−1 2 θ sin θ2

,

and VN (cos θ, sin θ) = =

− sin N θ + sin(N + 1)θ − sin(N + n)θ + sin(N + n − 1)θ 2(1 − cos θ) cos 2N2+1 θ − cos 2N +2n−1 θ 2 2 sin θ2

=

sin(N + n2 )θ sin n−1 2 θ sin θ2

.

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Main Results (n+1)

From the study in Section 3, we know that the Dirichlet kernels DN (x) on S n depend (n+1) −1 only on the angle θ, where θ = arccos x0 , θ ∈ [0, π], which means that DN (y x) on S n depend only on Re(y −1 x). In addition, when x, y ∈ S n , we have that Re(y −1 x) = Re(¯ y x) = n

y, x. Now, let x be a fixed point on S and write y = x cos θ + y˜ sin θ, where y˜ is orthogonal to x and θ = arg(y, x). In this case, y −1 = y¯ = x ¯ cos θ + y¯˜ sin θ and Re(y −1 x) = y, x = cos θ. We take average of a function f ∈ L2 (S n ) over the (n − 1)-dimensional sphere whose points y satisfy arg(y, x) = θ. This average is denoted by  1 Φx (f )(θ) = f (x cos θ + y˜ sin θ)dσn−1 (˜ y ), ωn−1 S n−1 where y˜ is the spherical variable on S n−1 , dσn−1 (˜ y ) is the normalized surface area measure on S n−1 . We call Φx (f )(θ) the average of f about x in angle θ. Note that in considering convergence problems it suffices to assume f to be scalar-valued. Without repeating, we will always assume this for the rest of the article. We briefly write Φx (f )(θ) = Φx (θ). In addition, we use W2l,1 ([0, π]) to denote the Sobolev space

 ∂ W2l,1 ([0, π]) = g ∈ L2 ([0, π])|( )k g ∈ L1 ([0, π]), k = 1, 2, · · · , l . ∂θ Denote Φx (0) = lim Φx (θ) if the limit exists. Then, the pointwise convergence theorem is θ→0

as follows. Theorem 4.1 Let x be a fixed point on S n . For any f ∈ L2 (S n ), we assume that Φx (0) exists. In addition, if Φx (θ) ∈ W2l,1 ([0, π]), then lim SN (f )(x) = Φx (0). If, in particular, f is continuous at x, then lim SN (f )(x) = f (x).

N →∞

N →∞

Proof It suffices to show lim (SN (f )(x) − Φx (0)) = 0. From the relations (2.3) and N →∞

(2.4), we have



1 ωn It is reduced to show 1 N →∞ ωn

(n+1)

Sn



DN

(n+1)

lim

Sn

DN

(y −1 x)dσ(y) = 1.

(y −1 x)(f (y) − Φx (0))dσ(y) = 0.

The last integral is scalar-valued and thus is induced from the scalar part of the N -th Dirichilet (n+1) (2l+2) kernel DN = DN , viz.  1 (2l+2) −1 D (y x)(f (y) − Φx (0))dσ(y). ωn S n N (2l+2)

Substituting the expression of DN the above becomes 1 ωn



π 0

(y −1 x) and writing the integral into an iterated integral,

j l 

 ∂ (j) (sin θ)(2l+2−2) κ−1 (2l)!! Clj (sin θ)j−2l Qi (sin θ, cos θ)( )i UN n ∂θ j=1 i=1

·(Φx (θ) − Φx (0))dθ

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π

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j l 

 ∂ (j) (sin θ)(2l+2−2) κ−1 Clj (sin θ)j−2l Ri (sin θ, cos θ)( )i VN n (2l)!! ∂θ j=1 i=1

·(Φx (θ) − Φx (0))dθ

 π j l   1 −1 ∂ (j) j = κ (2l)!! Cl ( )i UN [(sin θ)j Qi (sin θ, cos θ)(Φx (θ) − Φx (0))]dθ ωn n ∂θ 0 j=1 i=1

 π j l   ∂ 1 −1 (j) j Cl ( )i VN [(sin θ)j Ri (sin θ, cos θ)(Φx (θ) − Φx (0))]dθ. + κn (2l)!! ωn ∂θ 0 j=1 i=1

Taking integration by parts repeatedly, we have lim (SN (f )(x) − Φx (0))

N →∞

 π j l   1 −1 ∂ i j = lim κn (2l)!! (−1) Cl UN (cos θ, sin θ)( )i N →∞ ωn ∂θ 0 j=1 i=1

(j) · (sin θ)j Qi (sin θ, cos θ)(Φx (θ) − Φx (0)) dθ  π j l   1 −1 ∂ κn (2l)!! (−1)i Clj VN (cos θ, sin θ)( )i N →∞ ωn ∂θ 0 j=1 i=1

(j) · (sin θ)j Ri (sin θ, cos θ)(Φx (θ) − Φx (0)) dθ + lim

= lim I1 + lim I2 . N →∞

N →∞

We first consider the second part of the above expression. Since Φx (θ) ∈ W2l,1 ([0, π]), Φx (θ) − (j) ∂ i Φx (0) belongs to W2l,1 ([0, π]). As a consequence, ( ∂θ ) [(sin θ)j Ri (sin θ, cos θ)(Φx (θ) − Φx (0))], 1 ≤ i ≤ j, are all integrable. Replacing VN by Riemann–Lebesgue lemma, we have lim I2 = 0.

N →∞ sin(N + n )θ cos

sin(N + n 2 )θ sin sin θ2 n−1

n−1 2 θ

and applying the classical

θ

2 2 , there will be a singular point θ = 0, As for I1 , since UN is equal to sin θ2 thus the classical Riemann–Lebesgue lemma cannot be directly used. However, for any j, j  1 ≤ j ≤ l, we first consider the second summation . There is a factor (sin θ)j in the

i=1

∂ ∂ i integrand and the order of ∂θ is from 1 to j. When 1 ≤ i ≤ j − 1, after taking ( ∂θ ) on (j) j m1 (sin θ) Qi (sin θ, cos θ)(Φx (θ) − Φx (0)), there must be a factor (sin θ) left, m1 ≥ 1. If i = (j) ∂ j j, after taking ( ∂θ ) on (sin θ)j Qi (sin θ, cos θ)(Φx (θ) − Φx (0)), there also must be a factor (j) (sin θ)m2 left, m2 ≥ 1, except the first item that is j!(cos θ)j Qi (sin θ, cos θ)(Φx (θ) − Φx (0)). By using the classical Riemann–Lebesgue lemma for I1 except the first items of the sumj  of I1 , we have mation i=1

 π l  1 −1 κn (2l)!! (−1)j Clj UN (cos θ, sin θ) N →∞ ωn 0 j=1

lim I1 = lim

N →∞

(j)

·[j!(cos θ)j Qj (sin θ, cos θ)(Φx (θ) − Φx (0))]dθ  π l  sin(N + n2 )θ cos n−1 1 −1 2 θ κn (2l)!! (−1)j j!Clj = lim θ N →∞ ωn sin 2 0 j=1

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(j)

·(cos θ)j Qj (sin θ, cos θ)(Φx (θ) − Φx (0))dθ  l 1  1 π sin((N + n−1 2π −1 2 ) + 2 )θ (j) κn (2l)!! (−1)j j!Clj lim Ψx (θ)dθ, = θ N →∞ π 0 ωn 2 sin 2 j=1 (j)

(j)

j where Ψx (θ) = cos n−1 2 θ(cos θ) Qj (sin θ, cos θ)(Φx (θ) − Φx (0)), 1 ≤ j ≤ l. Since Φx (θ) ∈ (j)

L2 ([0, π]), then, obviously, Ψx (θ) ∈ L2 ([0, π]), 1 ≤ j ≤ l. By the classical Carleson’s theorem, we obtain l  2π −1 lim I1 = κn (2l)!! (−1)j j!Clj Ψx(j) (0) = 0. N →∞ ωn j=1 Therefore, we have lim SN (f )(x) = Φx (0). N →∞

Furthermore, by the result of R.A. Hunt ([7]), we can obviously extend the main convergence results to Lp cases, 1 < p < ∞. Now, we illustrate the results of Theorem 4.1 by two examples. Example 4.1 It should be observed that for n + 1 = 2, i.e., l = 0 in Theorem 4.1, the average Φx (θ) becomes the simple evaluation at two endpoints of the interval (−θ, θ), Φx (θ) =

1 [f (θx + θ) + f (θx − θ)], 2 (2)

where θx is the angle between x and e1 , and the Dirichlet kernel DN is just equal to DN in Section 1. From the condition of Theorem 4.1, Φx ∈ L2 ([0, π]) if and only if 12 [f (θx + θ) + f (θx − θ)] ∈ L2 ([0, π]). In particular, if x = 1, Theorem 4.1 reduces to the classical Carleson’s theorem. Example 4.2 Let l = 1, i.e., n + 1 = 4. Theorem 4.1 becomes the following. Let q be a fixed point on S 3 . For any f ∈ L2 (S 3 ), we assume that Φq (0) exists. In addition, if Φq (θ) ∈ L2 ([0, π]) and its derivative Ψq (θ) exists and Ψq (θ) is integrable in (0, π), then lim SN (f )(q) = Φq (0). Furthermore, lim SN (f )(q) = f (q) if f is continuous at q. N →∞

N →∞

The convergence results in [9], with the assumption that Φq (θ) is absolutely continuous in [0, π], is a consequence of the above results. In fact, if Φq (θ) is absolutely continuous in [0, π], then Φq (θ) ∈ L2 ([0, π]) and its derivative exists almost everywhere and is integrable on [0, π]. For l ≥ 1 the condition in Theorem 4.1 may be further weakened to the following: ⎧ n−1 j (j) 2 ⎪ ⎪ ⎪ (i) cos 2 θ(cos θ) Qj (sin θ, cos θ)(Φx (θ) − Φx (0)) ∈ L ([0, π]), 1 ≤ j ≤ l; ⎪ ⎪ ⎨ ∂ (j) (ii) ( )i [(sin θ)j Qi (sin θ, cos θ)(Φx (θ) − Φx (0))] ∈ L1 ([0, π]), 1 ≤ i ≤ j ≤ l; ∂θ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ (iii) ( ∂ )i [(sin θ)j Ri(j) (sin θ, cos θ)(Φx (θ) − Φx (0))] ∈ L1 ([0, π]), 1 ≤ i ≤ j ≤ l. ∂θ Obviously, Φx (θ) ∈ W2l,1 ([0, π]) implies the above conditions. The main results of [17] then can be easily deduced. In fact, there holds Corollary 4.1 If f is differentiable up to order l on S n , l ≥ 1, i.e., f ∈ C l (S n ), then lim SN (f )(x) = f (x),

N →∞

∀x ∈ S n .

Proof By Whitney’s extension theorem ([19], also see [17]), we can show that Φx (θ) ∈ ∂ i C ((0, π)), and (sin θ)j ( ∂θ ) Φx (θ), 1 ≤ i ≤ j ≤ l, converge to zero at 0 and π. In particular, l

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ACTA MATHEMATICA SCIENTIA

Vol.29 Ser.B

∂ i Φx (θ) is continuous on [0, π]. Thus, we have that Φx (θ) ∈ L2 ([0, π]) and ( ∂θ ) [(sin θ)j Φx (θ)] 1 ≤ i ≤ j ≤ l, are integrable in [0, π]. Then, the conditions (i), (ii), and (iii) are satisfied since (j) (j) Qi and Ri are polynomials in sin θ and cos θ. In addition, f is continuous on S n , then we have, lim SN (f )(x) = f (x), ∀x ∈ S n . N →∞

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