- Email: [email protected]
On non-compactly supported p-adic wavelets
J. Math. Anal. Appl. 443 (2016) 1260–1266
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On non-compactly supported p-adic wavelets ✩ S. Evdokimov a,b,∗ a
St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Fontanka-27, 191023 St. Petersburg, Russia b St. Petersburg State University, Universitetskii pr.-35, 198504 St. Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 5 February 2016 Available online 1 June 2016 Submitted by E. Saksman Keywords: p-Adic field Orthonormal wavelet basis Non-compactly supported vector-function
a b s t r a c t For any prime p, we prove the existence of non-compactly supported orthogonal p-adic wavelet bases in the Hilbert space L2 (Qp ), and construct the first explicit example of such a basis. The reasons are based on a special parametrization of the set of eigen standard Haar vector-functions. It should be noted that all previously known orthogonal p-adic wavelet bases were modifications of the p-adic Haar basis. © 2016 Elsevier Inc. All rights reserved.
1. Introduction A significant interest to p-adic wavelets and their applications emerged in the last decade (see [1–5,8, 10–12], review [9] and references within). The first p-adic wavelet basis was constructed by Kozyrev [7]. This basis can be regarded as a p-adic analog of the classical Haar basis. A number of constructions of p-adic wavelet bases and frames appeared later. All these wavelet systems consisted of so-called test functions, i.e. compactly supported functions with compactly supported Fourier transforms. However, it was proved in [6] that any orthogonal p-adic wavelet basis consisting of test functions is a kind of modification of the p-adic Haar basis obtained in [7]. Thus, up to now it has not been known whether there is an orthogonal p-adic basis which is not a “damaged Haar basis”, in contrast to the wavelet theories on other structure such as the Cantor/Vilenkin group and fields of positive characteristic. In this paper we give an explicit construction of orthogonal p-adic wavelet basis which is essentially different from the Haar basis. Let p be a prime, Qp the field of p-adic numbers and χp be the additive p-adic character. In paper [6] it was proved that an arbitrary periodic vector-function generating an orthonormal wavelet basis (ONWB in ✩
This research was supported by Grant 15-01-05796 of RFBR.
* Correspondence to: St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Fontanka-27, 191023 St. Petersburg, Russia. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2016.05.067 0022-247X/© 2016 Elsevier Inc. All rights reserved.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
1261
the sequel) in the Hilbert space L2 (Qp ), is wavelet equivalent in the sense of that paper to a vector-function (ν) Ψ = (ψ (1) , . . . , ψ (p−1) )T such that ψ (ν) ∈ W0 for all ν where (ν)
W0
= {f ∈ W0 : f (x + 1) = χp
ν f (x), x ∈ Qp } p
is the νth eigen subspace of the p-adic Haar wavelet space W0 . Such vector-functions were called eigen standard Haar ones in [6]. Moreover, if Ψ is compactly supported (this means that the function ψ (ν) is compactly supported for all ν), then Ψ is wavelet equivalent to the basic Haar vector-function Θ = (θ(1) , . . . , θ(p−1) )T where θ(ν) (x) = ϕ(x)χp
νx , p
ν = 1, . . . , p − 1,
with ϕ the characteristic function of the ring Zp of p-adic integers. The latter vector-function generates the p-adic ONWB found in [7]. On the other hand, nothing can be said when the vector-function Ψ is not compactly supported. Moreover, up to now no non-compactly supported vector-function generating either an ONWB, or wavelet Riesz basis, or wavelet frame has been found. The aim of the present paper is to fill this gap. Namely, we prove the existence of such a vector-function (Corollary 2) and also for each p provide an explicit example of it (Theorem 3). The reasons are based on the following parametrization of the set of eigen standard Haar vector-functions. Theorem 1. There exists a 1–1 correspondence between the set of eigen standard Haar vector-functions Ψ, and the set M p−1 where M = {g ∈ L2 (Zp ) : |g(x)| = 1 for all x ∈ Zp }. Moreover, the compactly supported vector-functions Ψ correspond to the periodic vector-functions belonging to M p−1 . Since, obviously, not every element of the set M is a periodic function, we come to the following statement. Corollary 2. For any prime p, there exists a non-compactly supported standard Haar vector-function. The following theorem provides an explicit form of Corollary 2. As it is shown in Section 3, the vectorfunction Ψ defined in the theorem corresponds to the tuple (g, . . . , g) ∈ M p−1 under the correspondence in × Theorem 1, where g is the function defined by the condition g(x) = (−1)n if x ∈ pn Z× p where Zp is the group of p-adic units. Below for a function f on Qp and a ∈ Qp , the function f0,a is defined by f0,a (x) = f (x − a). Theorem 3. For a prime p and each ν = 1, . . . , p − 1 let us define a function ψ (ν) on Qp by ψ (ν) =
f (a)χp
a∈Ip
νa (ν) θ p 0,a
where Ip = pkn ∈ Qp : k = 0, . . . , pn − 1, n = 0, 1, . . . , χp is the additive p-adic character, θ(ν) is the νth basic Haar function and f (a) =
p−1 p+1 , 2(−1)n pn−1 (p+1) ,
if a = 0; if a =
k pn ,
k not divisible by p.
Then Ψ := (ψ (1) , . . . , ψ (p−1) )T is a non-compactly supported standard Haar vector-function.
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The paper is organized as follows. Theorems 1 and 3 are proved in Section 3. Notation and basic facts on p-adic analysis and the p-adic wavelet theory we need, are concentrated in Section 2. 2. Notations and basic facts Here and in what follows, we use the notation and results from [13]. As for the p-adic wavelet theory and the p-adic multiresolution analysis (MRA in the sequel) theory, we refer to [2,6,12]. As usual, Z and Q denote the ring of rational integers and the field of all rational numbers, respectively. The field Qp of p-adic numbers is the completion of the field Q with respect to the p-adic norm | · |p defined as follows: |x|p =
p−γ , 0,
if x = pγ m n = 0; if x = 0,
where γ ∈ Z and m, n are integers not divisible by p. The extension of this norm to Qp is also denoted by | · |p . The norm | · |p is non-Archimedean, i.e. satisfies the strong triangle inequality |x + y|p ≤ max(|x|p , |y|p ),
x, y ∈ Qp .
Any p-adic number x = 0 can uniquely be written in the form x=
∞
xj p j
(2.1)
j=γ
where γ ∈ Z and xj ∈ {0, 1, . . . , p − 1} with xγ = 0. The fractional part {x}p of the number x equals by −1 definition j=γ xj pj . We also set {0}p = 0. The ring of p-adic integers Zp and the group of p-adic units are defined by Zp = {x ∈ Qp : |x|p ≤ 1},
Z× p = {x ∈ Qp : |x|p = 1}.
We observe that Z× p equals the multiplicative group of the ring Zp and Zp = {x ∈ Qp : {x}p = 0},
Zp \ {0} =
∞
pn Z× p.
n=0
The set Ip of representatives for the factor group Qp /Zp is defined by Ip =
k n ∈ Q : k = 0, . . . , p − 1, n = 0, 1 . . . . p pn
We observe that Ip = {x ∈ Qp : {x}p = x}, and the translations of Zp by elements of Ip are mutually disjoint and the union of them equals Qp . The additive character χp of the field Qp (or the additive p-adic character) is defined by χp (x) = e2πi{x}p ,
x ∈ Qp .
The field Qp is locally compact. Denote by dx the normalized Haar measure on it. By definition this measure is positive, invariant under translations and satisfies the condition Zp dx = 1. Moreover,
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
d(ax) = |a|p dx,
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a ∈ Qp \ {0}.
The Hilbert space of all complex-valued functions on Qp square integrable with respect to the measure dx, is denoted by L2 (Qp ). The inner product in this space is given by
f, gL2 (Qp ) =
f (x)g(x) dx Qp
where f, g ∈ L2 (Qp ). The Fourier transform of a locally-constant compactly supported function f on Qp (so-called test function) is defined as f(ξ) =
χp (ξx)f (x) dx,
ξ ∈ Qp .
Qp
By the Plancherel theorem the mapping f → f can uniquely be extended to a unitary isomorphism of L2 (Qp ). Moreover, (f)(x) = f (−x) identically on Qp for all f ∈ L2 (Qp ). The restrictions of the above isomorphism of L2 (Qp ) to the space of functions constant on cosets of Zp and to the space of functions supported on Zp induce unitary isomorphisms (Fourier transforms) L2 (Qp /Zp ) → L2 (Zp )
and L2 (Zp ) → L2 (Qp /Zp ),
respectively. Here the Haar measures on the discrete group Qp /Zp and compact group Zp are induced from Qp . In particular, the measure on the group Qp /Zp is a counting one. Given an integer m ∈ Z, a function f ∈ L2 (Qp ) is said to be pm -periodic (or pm -locally constant) if f (x) = f (y) for any x, y ∈ Qp with |x − y|p ≤ p−m (equivalently, f (x + pm ) = f (x), x ∈ Qp ). From the definition of Fourier transform it follows that f is pm -periodic
f is supported on p−m Zp .
⇐⇒
(2.2)
The function f is said to be periodic if it is pm -periodic for some m. Below for f ∈ L2 (Qp ), m ∈ Z and a ∈ Qp , we set fm,a (x) = p
m/2
f
x −a , pm
x ∈ Qp .
(2.3)
Let ϕ be the characteristic function of the set Zp . For every integer m, the functions ϕm,a , a ∈ Ip , form an orthonormal system. The sampling space Vm of the p-adic Haar MRA is defined by Vm = span {ϕm,a : a ∈ Ip }.
(2.4)
Any function ϕm,a is pm -periodic. Moreover, Vm = {f ∈ L2 (Qp ) : f is pm -periodic}. The union of all spaces Vm is dense in L2 (Qp ), their intersection is trivial and Vm ⊂ Vm+1 for all m ∈ Z.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
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The wavelet space Wm of the p-adic Haar MRA is defined as the orthogonal complement of Vm in Vm+1 Wm = Vm+1 Vm .
(2.5)
Then Wm = {f1,0 : f ∈ Wm−1 } for all m ∈ Z and we have the following orthogonal decomposition L2 (Qp ) =
Wm .
(2.6)
m∈Z
Moreover, Wm =
p−1
(ν) Wm ,
(2.7)
ν=1
(ν) where Wm = {f ∈ Wm : f0,pm = χp − νp f } is the νth eigen subspace of the p-adic Haar wavelet space Wm . Let Ψ = (ψ (1) , . . . , ψ (r) )T where ψ (ν) ∈ L2 (Qp ) for all ν = 1, . . . , r. We say that the vector-function Ψ generates an orthonormal wavelet basis (ONWB) if the system of functions (ν) {ψm,a : m ∈ Z, a ∈ Ip , ν = 1, . . . , r}
is an orthonormal basis for L2 (Qp ). The number r is called the rank of Ψ. A vector-function generating an ONWB is called standard Haar if it is of rank p −1 and all its components are in W0 . The ONWB is called standard Haar basis in this case. A standard Haar vector-function Ψ = (ν) (ψ (1) , . . . , ψ (p−1) )T is called eigen if ψ (ν) ∈ W0 for all ν = 1, . . . , p − 1. 3. Proof of main results Proof of Theorem 1. The theorem immediately follows from Lemmas 4, 5 and 6 below. 3 (ν)
Lemma 4. Let Ψ = (ψ (1) , . . . , ψ (p−1) )T be a vector-function with ψ (ν) ∈ W0 for all ν. Then Ψ is a standard (ν) (ν) Haar vector-function if and only if for each ν the set {ψ0,b : b ∈ Ip } is an orthonormal basis in W0 . (ν)
Proof. Since the wavelet space W0 equals the orthogonal sum of the spaces W0 , ν = 1, . . . , p −1 (see (2.7)), the lemma follows from the fact that the family of spaces {Vn }n∈Z forms an orthogonal MRA for the space L2 (Qp ) (see (2.6)). 3 For a function f ∈ L2 (Qp /Zp ) set ψ (ν) =
a∈Ip
f (a)χp (
νa (ν) )θ , p 0,a
where a is the class modulo Zp containing a. For each b ∈ Qp /Zp define a function fb on Qp /Zp by fb (a) = f (a − b). Lemma 5. For each ν = 1, . . . , p − 1 the mapping f → ψ (ν) induces a unitary isomorphism from L2 (Qp /Zp ) (ν) onto W0 . Moreover, given f ∈ L2 (Qp /Zp ) we have (ν)
(ν)
(1) the set {ψ0,b : b ∈ Ip } is an orthonormal basis in W0 if and only if the set {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ), (2) ψ (ν) is compactly supported if and only if f is finitely supported.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
1265
(ν)
Proof. The first part of the lemma easily follows from the fact that the set {θ0,a : a ∈ Ip } is an orthonormal (ν)
basis in the space W0 . Now let f ∈ L2 (Qp /Zp ). A straightforward computation shows that the function (ν) χp ( νa p )θ0,a does not depend on the choice of a in a given class modulo Zp . This implies that for each b ∈ Qp , the function fb goes to (below a = b + c + na where c ∈ Ip and na ∈ Z)
f (a − b)χp
a∈Ip
νa (ν) ν(b + c + na ) (ν) θ0,a = θ0,b+c+na f (c)χp p p c∈Ip
=
f (c)χp
c∈Ip
ν(b + c) (ν) νb νb (ν) νc (ν) θ0,b+c = χp ψ , f (c)χp ( )θ0,c 0,b = χp p p p p 0,b c∈Ip
(ν)
whence statement (1) follows. To prove statement (2) it suffices to note that every function θ0,a is compactly supported. 3 For a function f ∈ L2 (Qp /Zp ) denote by f ∈ L2 (Zp ) the Fourier transform of f . Lemma 6. The mapping f → f induces a unitary isomorphism from L2 (Qp /Zp ) onto L2 (Zp ). Moreover, given f ∈ L2 (Qp /Zp ) we have (1) the set {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ) if and only if |f(x)| = 1 for all x ∈ Zp , (2) f is finitely supported if and only if f is periodic. Proof. The first part of the lemma follows from the Plancherel theorem. To prove statement(1) we observe that since for every a ∈ Qp /Zp the Fourier transform of the function fa equals fχap where by definition χap (x) = χp (ax), x ∈ Zp , we have fa , fb L2 (Qp /Zp )
= fχap , fχpb L2 (Zp ) =
f(x)χap (x) f(x)χpb (x) dx =
Zp
2 χa−b p (x) |f (x)| dx.
Zp
So, if the set Sf := {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ), then the function |f|2 is orthogonal to any non-trivial character belonging to the set {χpb : b ∈ Qp /Zp } of all characters of Zp , and Zp |f(x)|2 dx = 1. However, the characters of Zp form an orthonormal basis of L2 (Zp ). Thus this function equals 1 for all x ∈ Zp . Conversely, if the function |f|2 equals 1 for all x ∈ Zp , then the set {f(x)χb (x) : b ∈ Qp /Zp } is an orthonormal basis in L2 (Zp ) and, consequently, the set Sf is an orthonormal p
basis in L2 (Qp /Zp ). Statement (2) follows from (2.2). 3 Proof of Theorem 3. By Lemmas 4, 5 and 6 it suffices to show that f (a) = g(a) for all a ∈ Ip where g is the Fourier transform of the function g ∈ L2 (Zp ) defined by the condition g(x) = (−1)n if x ∈ pn Z× p. If a = 0, then we have
g(0) =
g(x) dx = Zp
∞ n=0
(−1)n
dx =
pn Z× p
∞
(−1)n
n=0
p−1 p−1 = f (a). = n+1 p p+1
To treat the case a = 0 we need the following statement. Lemma 7. Let a = k/pm where m ≥ 1 and k is not divided by p. Then
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
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pn Z× p
⎧ 1 ⎪ ⎪ ⎨− pm , χap (x) dx = pp−1 n+1 , ⎪ ⎪ ⎩0,
if n = m − 1; if n > m − 1; if n < m − 1.
Proof. If n > m − 1, then χap (x) = 1 on pn Z× , so that the left-hand side equals pn Z× dx = (p − 1)/pn+1 . p n × m−1 If n < m − 1, then the set p Zp is the union of cosets of the group p Zp . Since the character χap is not identical 1 on the latter group, the left-hand side equals 0. If n = m − 1, then the set pn Z× p is the union of p − 1 cosets pm−1 i + pm Zp , i = 1, . . . , p − 1 of the group pm Zp . Thus the left-hand side equals p−1
χap (pm−1 i)
i=1
dx =
pm Z
p
p−1 1 1 χp (ki/p) = − m . m p i=1 p
3
Now, if a ∈ Ip \ {0}, then a = k/pm with m > 0 and k not divided by p, and by Lemma 7 we have
g(a) =
g(x)χap (x) dx Zp
=
=
∞
n
(−1)
n=0
χap (x) dx
pn Z×
∞ 2(−1)m (−1)m (−1)m (−1)m (p − 1) (−1)n (p − 1) = = f (a). + = + pm pn+1 pm pm (p + 1) pm−1 (p + 1) n=m
3
References [1] S. Albeverio, S. Evdokimov, M. Skopina, p-Adic nonorthogonal wavelet bases, Proc. Steklov Inst. Math. 265 (2009) 1–12. [2] S. Albeverio, S. Evdokimov, M. Skopina, p-Adic multiresolution analysis and wavelet frames, J. Fourier Anal. Appl. 16 (5) (2010) 693–714. [3] J.J. Benedetto, R.L. Benedetto, A wavelet theory for local fields and related groups, J. Geom. Anal. 3 (2004) 423–456. [4] S. Evdokimov, Haar multiresolution analysis and Haar bases on the ring of rational adeles, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 400 (2012) 158–165 (in Russian); translation: J. Math. Sci. (N. Y.) 192 (2013) 215–219. [5] S.A. Evdokimov, M.A. Skopina, 2-Adic wavelet bases, Proc. Inst. Math. Mech., Ural Branch Russian Acad. Sci. 15 (1) (2009) 135–146 (in Russian). [6] S. Evdokimov, M. Skopina, On orthogonal p-adic wavelet bases, J. Math. Anal. Appl. 424 (2015) 952–965. [7] S.V. Kozyrev, Wavelet analysis as a p-adic spectral analysis, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2) (2002) 149–158. [8] S.V. Kozyrev, p-Adic pseudodifferential operators and p-adic wavelets, Theoret. Math. Phys. 138 (3) (2004) 322–332. [9] S.V. Kozyrev, A.Yu. Khrennikov, V.M. Shelkovich, p-Adic wavelets and their applications, Proc. Steklov Inst. Math. 285 (2014) 157–196. [10] S.F. Lukomskii, Haar system on the product of groups of p-adic integers, Math. Notes 90 (3–4) (2011) 517–532. [11] V.M. Shelkovich, p-Adic evolution pseudo-differential equations and p-adic wavelets, Izv. Math. 75 (3) (2011) 1249–1278. [12] V.M. Shelkovich, M. Skopina, p-Adic Haar multiresolution analysis and pseudo-differential operators, J. Fourier Anal. Appl. 15 (3) (2009) 366–393. [13] V.S. Vladimirov, I.V. Volovich, E.I. Zelenov, p-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.
Contents lists available at ScienceDirect
Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa
On non-compactly supported p-adic wavelets ✩ S. Evdokimov a,b,∗ a
St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Fontanka-27, 191023 St. Petersburg, Russia b St. Petersburg State University, Universitetskii pr.-35, 198504 St. Petersburg, Russia
a r t i c l e
i n f o
Article history: Received 5 February 2016 Available online 1 June 2016 Submitted by E. Saksman Keywords: p-Adic field Orthonormal wavelet basis Non-compactly supported vector-function
a b s t r a c t For any prime p, we prove the existence of non-compactly supported orthogonal p-adic wavelet bases in the Hilbert space L2 (Qp ), and construct the first explicit example of such a basis. The reasons are based on a special parametrization of the set of eigen standard Haar vector-functions. It should be noted that all previously known orthogonal p-adic wavelet bases were modifications of the p-adic Haar basis. © 2016 Elsevier Inc. All rights reserved.
1. Introduction A significant interest to p-adic wavelets and their applications emerged in the last decade (see [1–5,8, 10–12], review [9] and references within). The first p-adic wavelet basis was constructed by Kozyrev [7]. This basis can be regarded as a p-adic analog of the classical Haar basis. A number of constructions of p-adic wavelet bases and frames appeared later. All these wavelet systems consisted of so-called test functions, i.e. compactly supported functions with compactly supported Fourier transforms. However, it was proved in [6] that any orthogonal p-adic wavelet basis consisting of test functions is a kind of modification of the p-adic Haar basis obtained in [7]. Thus, up to now it has not been known whether there is an orthogonal p-adic basis which is not a “damaged Haar basis”, in contrast to the wavelet theories on other structure such as the Cantor/Vilenkin group and fields of positive characteristic. In this paper we give an explicit construction of orthogonal p-adic wavelet basis which is essentially different from the Haar basis. Let p be a prime, Qp the field of p-adic numbers and χp be the additive p-adic character. In paper [6] it was proved that an arbitrary periodic vector-function generating an orthonormal wavelet basis (ONWB in ✩
This research was supported by Grant 15-01-05796 of RFBR.
* Correspondence to: St. Petersburg Department of Steklov Institute of Mathematics, St. Petersburg, Fontanka-27, 191023 St. Petersburg, Russia. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmaa.2016.05.067 0022-247X/© 2016 Elsevier Inc. All rights reserved.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
1261
the sequel) in the Hilbert space L2 (Qp ), is wavelet equivalent in the sense of that paper to a vector-function (ν) Ψ = (ψ (1) , . . . , ψ (p−1) )T such that ψ (ν) ∈ W0 for all ν where (ν)
W0
= {f ∈ W0 : f (x + 1) = χp
ν f (x), x ∈ Qp } p
is the νth eigen subspace of the p-adic Haar wavelet space W0 . Such vector-functions were called eigen standard Haar ones in [6]. Moreover, if Ψ is compactly supported (this means that the function ψ (ν) is compactly supported for all ν), then Ψ is wavelet equivalent to the basic Haar vector-function Θ = (θ(1) , . . . , θ(p−1) )T where θ(ν) (x) = ϕ(x)χp
νx , p
ν = 1, . . . , p − 1,
with ϕ the characteristic function of the ring Zp of p-adic integers. The latter vector-function generates the p-adic ONWB found in [7]. On the other hand, nothing can be said when the vector-function Ψ is not compactly supported. Moreover, up to now no non-compactly supported vector-function generating either an ONWB, or wavelet Riesz basis, or wavelet frame has been found. The aim of the present paper is to fill this gap. Namely, we prove the existence of such a vector-function (Corollary 2) and also for each p provide an explicit example of it (Theorem 3). The reasons are based on the following parametrization of the set of eigen standard Haar vector-functions. Theorem 1. There exists a 1–1 correspondence between the set of eigen standard Haar vector-functions Ψ, and the set M p−1 where M = {g ∈ L2 (Zp ) : |g(x)| = 1 for all x ∈ Zp }. Moreover, the compactly supported vector-functions Ψ correspond to the periodic vector-functions belonging to M p−1 . Since, obviously, not every element of the set M is a periodic function, we come to the following statement. Corollary 2. For any prime p, there exists a non-compactly supported standard Haar vector-function. The following theorem provides an explicit form of Corollary 2. As it is shown in Section 3, the vectorfunction Ψ defined in the theorem corresponds to the tuple (g, . . . , g) ∈ M p−1 under the correspondence in × Theorem 1, where g is the function defined by the condition g(x) = (−1)n if x ∈ pn Z× p where Zp is the group of p-adic units. Below for a function f on Qp and a ∈ Qp , the function f0,a is defined by f0,a (x) = f (x − a). Theorem 3. For a prime p and each ν = 1, . . . , p − 1 let us define a function ψ (ν) on Qp by ψ (ν) =
f (a)χp
a∈Ip
νa (ν) θ p 0,a
where Ip = pkn ∈ Qp : k = 0, . . . , pn − 1, n = 0, 1, . . . , χp is the additive p-adic character, θ(ν) is the νth basic Haar function and f (a) =
p−1 p+1 , 2(−1)n pn−1 (p+1) ,
if a = 0; if a =
k pn ,
k not divisible by p.
Then Ψ := (ψ (1) , . . . , ψ (p−1) )T is a non-compactly supported standard Haar vector-function.
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The paper is organized as follows. Theorems 1 and 3 are proved in Section 3. Notation and basic facts on p-adic analysis and the p-adic wavelet theory we need, are concentrated in Section 2. 2. Notations and basic facts Here and in what follows, we use the notation and results from [13]. As for the p-adic wavelet theory and the p-adic multiresolution analysis (MRA in the sequel) theory, we refer to [2,6,12]. As usual, Z and Q denote the ring of rational integers and the field of all rational numbers, respectively. The field Qp of p-adic numbers is the completion of the field Q with respect to the p-adic norm | · |p defined as follows: |x|p =
p−γ , 0,
if x = pγ m n = 0; if x = 0,
where γ ∈ Z and m, n are integers not divisible by p. The extension of this norm to Qp is also denoted by | · |p . The norm | · |p is non-Archimedean, i.e. satisfies the strong triangle inequality |x + y|p ≤ max(|x|p , |y|p ),
x, y ∈ Qp .
Any p-adic number x = 0 can uniquely be written in the form x=
∞
xj p j
(2.1)
j=γ
where γ ∈ Z and xj ∈ {0, 1, . . . , p − 1} with xγ = 0. The fractional part {x}p of the number x equals by −1 definition j=γ xj pj . We also set {0}p = 0. The ring of p-adic integers Zp and the group of p-adic units are defined by Zp = {x ∈ Qp : |x|p ≤ 1},
Z× p = {x ∈ Qp : |x|p = 1}.
We observe that Z× p equals the multiplicative group of the ring Zp and Zp = {x ∈ Qp : {x}p = 0},
Zp \ {0} =
∞
pn Z× p.
n=0
The set Ip of representatives for the factor group Qp /Zp is defined by Ip =
k n ∈ Q : k = 0, . . . , p − 1, n = 0, 1 . . . . p pn
We observe that Ip = {x ∈ Qp : {x}p = x}, and the translations of Zp by elements of Ip are mutually disjoint and the union of them equals Qp . The additive character χp of the field Qp (or the additive p-adic character) is defined by χp (x) = e2πi{x}p ,
x ∈ Qp .
The field Qp is locally compact. Denote by dx the normalized Haar measure on it. By definition this measure is positive, invariant under translations and satisfies the condition Zp dx = 1. Moreover,
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
d(ax) = |a|p dx,
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a ∈ Qp \ {0}.
The Hilbert space of all complex-valued functions on Qp square integrable with respect to the measure dx, is denoted by L2 (Qp ). The inner product in this space is given by
f, gL2 (Qp ) =
f (x)g(x) dx Qp
where f, g ∈ L2 (Qp ). The Fourier transform of a locally-constant compactly supported function f on Qp (so-called test function) is defined as f(ξ) =
χp (ξx)f (x) dx,
ξ ∈ Qp .
Qp
By the Plancherel theorem the mapping f → f can uniquely be extended to a unitary isomorphism of L2 (Qp ). Moreover, (f)(x) = f (−x) identically on Qp for all f ∈ L2 (Qp ). The restrictions of the above isomorphism of L2 (Qp ) to the space of functions constant on cosets of Zp and to the space of functions supported on Zp induce unitary isomorphisms (Fourier transforms) L2 (Qp /Zp ) → L2 (Zp )
and L2 (Zp ) → L2 (Qp /Zp ),
respectively. Here the Haar measures on the discrete group Qp /Zp and compact group Zp are induced from Qp . In particular, the measure on the group Qp /Zp is a counting one. Given an integer m ∈ Z, a function f ∈ L2 (Qp ) is said to be pm -periodic (or pm -locally constant) if f (x) = f (y) for any x, y ∈ Qp with |x − y|p ≤ p−m (equivalently, f (x + pm ) = f (x), x ∈ Qp ). From the definition of Fourier transform it follows that f is pm -periodic
f is supported on p−m Zp .
⇐⇒
(2.2)
The function f is said to be periodic if it is pm -periodic for some m. Below for f ∈ L2 (Qp ), m ∈ Z and a ∈ Qp , we set fm,a (x) = p
m/2
f
x −a , pm
x ∈ Qp .
(2.3)
Let ϕ be the characteristic function of the set Zp . For every integer m, the functions ϕm,a , a ∈ Ip , form an orthonormal system. The sampling space Vm of the p-adic Haar MRA is defined by Vm = span {ϕm,a : a ∈ Ip }.
(2.4)
Any function ϕm,a is pm -periodic. Moreover, Vm = {f ∈ L2 (Qp ) : f is pm -periodic}. The union of all spaces Vm is dense in L2 (Qp ), their intersection is trivial and Vm ⊂ Vm+1 for all m ∈ Z.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
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The wavelet space Wm of the p-adic Haar MRA is defined as the orthogonal complement of Vm in Vm+1 Wm = Vm+1 Vm .
(2.5)
Then Wm = {f1,0 : f ∈ Wm−1 } for all m ∈ Z and we have the following orthogonal decomposition L2 (Qp ) =
Wm .
(2.6)
m∈Z
Moreover, Wm =
p−1
(ν) Wm ,
(2.7)
ν=1
(ν) where Wm = {f ∈ Wm : f0,pm = χp − νp f } is the νth eigen subspace of the p-adic Haar wavelet space Wm . Let Ψ = (ψ (1) , . . . , ψ (r) )T where ψ (ν) ∈ L2 (Qp ) for all ν = 1, . . . , r. We say that the vector-function Ψ generates an orthonormal wavelet basis (ONWB) if the system of functions (ν) {ψm,a : m ∈ Z, a ∈ Ip , ν = 1, . . . , r}
is an orthonormal basis for L2 (Qp ). The number r is called the rank of Ψ. A vector-function generating an ONWB is called standard Haar if it is of rank p −1 and all its components are in W0 . The ONWB is called standard Haar basis in this case. A standard Haar vector-function Ψ = (ν) (ψ (1) , . . . , ψ (p−1) )T is called eigen if ψ (ν) ∈ W0 for all ν = 1, . . . , p − 1. 3. Proof of main results Proof of Theorem 1. The theorem immediately follows from Lemmas 4, 5 and 6 below. 3 (ν)
Lemma 4. Let Ψ = (ψ (1) , . . . , ψ (p−1) )T be a vector-function with ψ (ν) ∈ W0 for all ν. Then Ψ is a standard (ν) (ν) Haar vector-function if and only if for each ν the set {ψ0,b : b ∈ Ip } is an orthonormal basis in W0 . (ν)
Proof. Since the wavelet space W0 equals the orthogonal sum of the spaces W0 , ν = 1, . . . , p −1 (see (2.7)), the lemma follows from the fact that the family of spaces {Vn }n∈Z forms an orthogonal MRA for the space L2 (Qp ) (see (2.6)). 3 For a function f ∈ L2 (Qp /Zp ) set ψ (ν) =
a∈Ip
f (a)χp (
νa (ν) )θ , p 0,a
where a is the class modulo Zp containing a. For each b ∈ Qp /Zp define a function fb on Qp /Zp by fb (a) = f (a − b). Lemma 5. For each ν = 1, . . . , p − 1 the mapping f → ψ (ν) induces a unitary isomorphism from L2 (Qp /Zp ) (ν) onto W0 . Moreover, given f ∈ L2 (Qp /Zp ) we have (ν)
(ν)
(1) the set {ψ0,b : b ∈ Ip } is an orthonormal basis in W0 if and only if the set {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ), (2) ψ (ν) is compactly supported if and only if f is finitely supported.
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
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(ν)
Proof. The first part of the lemma easily follows from the fact that the set {θ0,a : a ∈ Ip } is an orthonormal (ν)
basis in the space W0 . Now let f ∈ L2 (Qp /Zp ). A straightforward computation shows that the function (ν) χp ( νa p )θ0,a does not depend on the choice of a in a given class modulo Zp . This implies that for each b ∈ Qp , the function fb goes to (below a = b + c + na where c ∈ Ip and na ∈ Z)
f (a − b)χp
a∈Ip
νa (ν) ν(b + c + na ) (ν) θ0,a = θ0,b+c+na f (c)χp p p c∈Ip
=
f (c)χp
c∈Ip
ν(b + c) (ν) νb νb (ν) νc (ν) θ0,b+c = χp ψ , f (c)χp ( )θ0,c 0,b = χp p p p p 0,b c∈Ip
(ν)
whence statement (1) follows. To prove statement (2) it suffices to note that every function θ0,a is compactly supported. 3 For a function f ∈ L2 (Qp /Zp ) denote by f ∈ L2 (Zp ) the Fourier transform of f . Lemma 6. The mapping f → f induces a unitary isomorphism from L2 (Qp /Zp ) onto L2 (Zp ). Moreover, given f ∈ L2 (Qp /Zp ) we have (1) the set {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ) if and only if |f(x)| = 1 for all x ∈ Zp , (2) f is finitely supported if and only if f is periodic. Proof. The first part of the lemma follows from the Plancherel theorem. To prove statement(1) we observe that since for every a ∈ Qp /Zp the Fourier transform of the function fa equals fχap where by definition χap (x) = χp (ax), x ∈ Zp , we have fa , fb L2 (Qp /Zp )
= fχap , fχpb L2 (Zp ) =
f(x)χap (x) f(x)χpb (x) dx =
Zp
2 χa−b p (x) |f (x)| dx.
Zp
So, if the set Sf := {fb : b ∈ Qp /Zp } is an orthonormal basis in L2 (Qp /Zp ), then the function |f|2 is orthogonal to any non-trivial character belonging to the set {χpb : b ∈ Qp /Zp } of all characters of Zp , and Zp |f(x)|2 dx = 1. However, the characters of Zp form an orthonormal basis of L2 (Zp ). Thus this function equals 1 for all x ∈ Zp . Conversely, if the function |f|2 equals 1 for all x ∈ Zp , then the set {f(x)χb (x) : b ∈ Qp /Zp } is an orthonormal basis in L2 (Zp ) and, consequently, the set Sf is an orthonormal p
basis in L2 (Qp /Zp ). Statement (2) follows from (2.2). 3 Proof of Theorem 3. By Lemmas 4, 5 and 6 it suffices to show that f (a) = g(a) for all a ∈ Ip where g is the Fourier transform of the function g ∈ L2 (Zp ) defined by the condition g(x) = (−1)n if x ∈ pn Z× p. If a = 0, then we have
g(0) =
g(x) dx = Zp
∞ n=0
(−1)n
dx =
pn Z× p
∞
(−1)n
n=0
p−1 p−1 = f (a). = n+1 p p+1
To treat the case a = 0 we need the following statement. Lemma 7. Let a = k/pm where m ≥ 1 and k is not divided by p. Then
S. Evdokimov / J. Math. Anal. Appl. 443 (2016) 1260–1266
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pn Z× p
⎧ 1 ⎪ ⎪ ⎨− pm , χap (x) dx = pp−1 n+1 , ⎪ ⎪ ⎩0,
if n = m − 1; if n > m − 1; if n < m − 1.
Proof. If n > m − 1, then χap (x) = 1 on pn Z× , so that the left-hand side equals pn Z× dx = (p − 1)/pn+1 . p n × m−1 If n < m − 1, then the set p Zp is the union of cosets of the group p Zp . Since the character χap is not identical 1 on the latter group, the left-hand side equals 0. If n = m − 1, then the set pn Z× p is the union of p − 1 cosets pm−1 i + pm Zp , i = 1, . . . , p − 1 of the group pm Zp . Thus the left-hand side equals p−1
χap (pm−1 i)
i=1
dx =
pm Z
p
p−1 1 1 χp (ki/p) = − m . m p i=1 p
3
Now, if a ∈ Ip \ {0}, then a = k/pm with m > 0 and k not divided by p, and by Lemma 7 we have
g(a) =
g(x)χap (x) dx Zp
=
=
∞
n
(−1)
n=0
χap (x) dx
pn Z×
∞ 2(−1)m (−1)m (−1)m (−1)m (p − 1) (−1)n (p − 1) = = f (a). + = + pm pn+1 pm pm (p + 1) pm−1 (p + 1) n=m
3
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