Haar system on a product of zero-dimensional compact groups

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1 Cent. Eur. J. Math DOI: /s Central European Journal of Mathematics Haar system on a product of zero-dimensional compact groups Research Article Sergei F. Lukomskii Department of Mathematics and Mechanics, Saratov State University, Saratov, Russia Received 2 August 200; accepted 7 January 20 Abstract: In this work, we study the problem of constructing Haar bases on a product of arbitrary compact zero-dimensional Abelian groups. A general scheme for the construction of Haar functions is given for arbitrary dimension. For dimension d = 2, we describe all Haar functions. MSC: 43A75, 43A77, 43A70, R56 Keywords: Compact zero-dimensional groups Characters Haar functions Wavelet bases Versita Sp. z o.o.. Introduction In 90, A. Haar [] introduced an orthogonal system of functions on the interval [0, ]. The functions of this system result from contractions and shifts of one function H x = χ [0, 2 x χ [,x. This system of functions is the first 2 known wavelet, called the Haar basis. In [0], B. Golubov defined a generalization of Haar functions on the interval [0,, so-called P-adic Haar functions, where P = p n n=0 is a sequence of prime numbers, and presented a number of convergence theorems for P-adic Haar series. We can write these functions in modern notation as H jmn+kx = rnx j q [0, mn x q where m 0 =, + = p n, r n x = exp 2πij [ j, x, j +, j = 0, p n. p n LukomskiiSF@info.sgu.ru 627

2 Haar system on a product of zero-dimensional compact groups The first examples of orthogonal wavelets on the Cantor dyadic group were constructed by W.C. Lang [6 8]. The investigation of wavelets on the Cantor dyadic group was continued by Yu.A. Farkov and V.Yu. Protasov [9, 23]. In [8], Farkov proposed a general method of constructing compactly supported orthogonal wavelets on a locally compact Vilenkin group G with a constant generating sequence. An orthonormal compactly supported p-adic wavelet basis in L 2 Q p was established by S.V. Kozyrev [5]. It consists of functions H j,n,k x = p n χ j p p p n x q Zp p n x q, 2 j = 0, p, n Z, and is an analog of the p-adic Haar basis of Golubov [0] in the following sense. If we consider the natural map θ : Q p [0, we obtain p-adic Haar basis of Golubov see Section 2, Remarks 2.4 and 2.5. J.J. Benedetto and R.L. Benedetto [5, 6] suggested a method for finding wavelet bases on locally compact Abelian groups. Instead of developing the MRA approach, they used the theory of wavelet sets. S. Albeverio, A.Yu. Khrennikov, V.M. Shelkovich, and M.A. Skopina [3, 4, 7, 22, 24] following S. Mallat developed MRA multiresolution analysis theory in L 2 Q p. They described a wide class of p-adic refinement equations generating p-adic multiresolution analysis, and suggested a method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory. The theory of p-adic wavelets may by found in [2]. In [22], multidimensional dyadic orthogonal bases for L 2 Q d 2 were constructed by means of the tensor product of one-dimensional MRA. Let us mention also that in [4] E.J. King and Skopina constructed a wavelet basis in L 2 Q 2 2 using the quincunx matrix as a dilation matrix. The wavelet system corresponding to this MRA is a basis for L 2 Q 2 2, but it cannot be obtained by means of the separable p-adic MRA. In [2, 3] an infinite family of compactly supported non-haar p-adic wavelet bases in L 2 Q d p was constructed and a criterion for a multidimensional p-adic wavelet to be an eigenfunction for a pseudo-differential operator was derived. The author in [9] constructed a Haar system H jmn+kx = m /2 n r j nx q Gn +qx, j =, p n, k = 0,, 3 on a compact zero-dimensional Abelian group G, + with a basic chain G n n=0 and basic sequence g n n. The functions r n in 3 are Rademacher functions. If G is a Vilenkin group we obtain the Haar system ; if G is a ring Z p we obtain the Haar system 2. The MRA theory in L 2 G was developed by the author in [20] for any locally-compact zero-dimensional Abelian group G satisfying the condition G n /G n+ = p. If the refinable function φ is a solution of the refinement equation p φx = β j φax jg and ˆφ = G0, then an orthonormal wavelet basis generated by this MRA is the Haar basis 3. In [2], it was proved that a zero-dimensional compact Abelian group G, + is a product Z d p if and only if pg n = g n+d. Using this fact, Haar bases for L 2 Z d p were constructed in [2]. In the present paper, we study the problem of constructing a class of wavelet bases of Haar-type on a product of arbitrary compact zero-dimensional Abelian groups. We will call them Haar bases. We prove that the product of zero-dimensional groups is a zero-dimensional group too and give a method of constructing a chain of subgroups that generates the topology on the product. Using these chains of subgroups, we give a general scheme for the construction of Haar functions on a product of compact zero-dimensional groups. In a particular case, namely for dimension d = 2, we describe all such Haar functions. It should be noted [] that there exists a natural map θ : G [0, ] under which θg n = [ 0, ]. Therefore any Haar system on the product G d is an orthonormal basis in L 2 [0, ] d. It follows that constructing 2-dimensional Haar bases may be of use in image analysis. 628

3 S.F. Lukomskii 2. Preliminaries We start with some basic notions and facts related to analysis on zero-dimensional groups. A more detailed account may be found in []. Let G, + be a compact zero-dimensional Abelian group which satisfies the second countability axiom. It is known that a topology on such a group can be given by a chain of open subgroups G = G 0 G... G n..., 4 such that n=0 G n = {0} and G n /G n+ = p n. We may assume that p n are prime numbers, since the chain of subgroups can be refined so that the factor groups G n /G n+ have prime order. We will refer to this chain G n n as the basic chain. Picking one element g n G n \ G n+ for each n N 0 = N {0} we can represent any element x G in the form x = a n g n, a n = 0, p n. 5 n=0 On the other hand, for any sequence a n n=0, a n = 0, p n, the series 5 converges and its sum is an element of G. We call the system g n n a basic system. If the operation + in the group G satisfies the condition p n g n = g n+, the group G, + is the group Z P of P-adic integers. Cosets G n+ + jg n pn form a cyclic group of prime order. Cosets G n + h, n N 0, h G, together with the empty set form a semiring N. Let m 0 =, + = p n, n 0. Define a measure µ on the semiring N as follows: µg n + g = µg n =. It can be extended from the semiring N onto the σ-algebra for example, by using Carathéodory s extension. As a result, we obtain a translation invariant measure µ, which agrees on the Borel sets with the Haar measure on G. With fx dµx we will denote the absolutely convergent integral of a function f on G with respect to the measure µ. G Let X be the group of characters of the group G. Denote by G n = {χ X : χx = for all x G n } the annulator of the subgroup G n. Each annulator G n is a group with respect to multiplication and the subgroups G n form an ascending sequence {} = G 0 G... G n..., with n=0 G n = X and G n+ /G n = p n. For each n N 0, choose a character r n G n+ \ G n. We call r n a Rademacher function. Any character χ X can be represented in the form χ = rn αn, α n = 0, p n ; n=0 note that the number of factors not equal to one is finite. Lemma 2.. Rademacher functions r n are constant on cosets G n+ +h. 2 r n Gn+ + jg n, j = 0, pn, are distinct roots of unity of order p n. 629

4 Haar system on a product of zero-dimensional compact groups Proof. If x G n+ + h then x = x n+ + a n g n + a n g n + + a 0 g 0, x n+ G n+. Therefore n n r n x = r n x n+ r n g k a k = r n g k a k. k=0 k=0 2 Let H n,j = G n+ + jg n. Since p n is prime, H n,j pn is a cyclic group of order p n. Therefore for all j 0, H n,j + H n,j + + H n,j = H n,0. }{{} p n Hence r n H n,j pn =. Since the group G n /G n+ has prime order p n, all roots r n H n,j are distinct. In [9], Haar functions were defined on any compact zero-dimensional group as follows: H 0 x, H lmn+kx = m /2 n r l nx q Gn +qx, l =, p n, k = 0,, 6 where k = a n + a n a m + a 0 m 0 and G q = a n g n + a n 2 g n a g + a 0 g 0. It was proved that the Haar system H n n=0 is a complete orthonormal system in L 2G. Remark 2.2. It follows from Lemma 2. and the definition that the Haar functions do not depend on the group operation. Remark 2.3. For construction of Haar functions we must first set a basic chain and a basic sequence. Remark 2.4. Let us discuss a relation between the Haar functions H jmn+k on a zero-dimensional group G, + and the Haar function of Golubov on the interval [0, ]. Let g n n be a basic sequence and x G. We can write x in the form Define the mapping θ : G [0, ] as follows: x = a n g n, a n = 0, p n. n=0 θx = k=0 a k m k+ = y. The representation y in this form is called the P-adic expansion of y. The mapping θ establishes the one-to-one correspondence between G and the so-called modified segment [0, ]. The modified segment [0, ] can be interpreted as the closed segment [0, ] in which all P-adic rational points k k are counted twice: the left point 0 corresponds to the infinite P-adic expansion k = a 0 m + a m a n + p n + + p n+ +2 +, 630

5 S.F. Lukomskii and the right point It is easy to see that [ θg n = 0, k + 0 corresponds to the finite expansion k = a 0 m + a m a n. ] 0, θ [ ν G n + α 0 g α n g n = + 0, ν + ] [ 0 = 0, ] 0 + ν, where ν = α 0 p p 2... p n + α p 2 p 3... p n + + α n 2 p n + α n. Therefore we can write Haar functions 6 on [0, ] in the form H lmn+kx = r l n x νmn [0, x ν. mn 0] It is the Haar system, up to a null set. Remark 2.5. If p n = p for all n N we can define a dilation operator A: G G as follows: Ax = a n g n when x = a n g n. n= Then we can write the Haar functions 6 in the form n= H lp n +kx = p n/2 r j 0 A n x q G0 A n x q. If G = Z p, then Ax = p x, χ p p x = r 0 x and we obtain the Haar function of Kozyrev. 3. Construction of basic chain and basic sequence Let G, + be a compact zero-dimensional group defined with a basic chain 4. We denote by G = G d = G G G the direct sum of d copies of G. The base of neighborhoods of zero in G d consists of all products G n G n2 G nd. We can take the chain of d-dimensional cubes G n G n G n = G d n as the base of neighborhood of zero in G d. Note that the chain G = G d 0 G d... G d n... 7 is not a basic chain, since G d n/g d n+ = p d n is not a prime number. Denote G d n = G nd and refine the chain 7 to obtain a basic chain in the following way. Let G d n+ Gd n. Take a nonzero element g n+d = a n 0 g n, a n g n,..., a n d g n Gnd \ G n+d. 63

6 Haar system on a product of zero-dimensional compact groups Cosets form a group of prime order p n, whence the set Gn+d + jg n+d pn G n+d = p n Gn+d + jg n+d, where stands for the disjoint union, is a group such that G n+d G n+d G nd, G n+d /G n+d = p n, and G nd /G n+d = p d n. If d >, take an element g n+d 2 = a 2 n 0 g n, a 2 n g n,..., a 2 n d g n Gnd \ G n+d. Similarly to the previous situation, we conclude that the set G n+d 2 = p n Gn+d + jg n+d 2 is a group such that G n+d G n+d G n+d 2 G nd, G n+d 2 /G n+d = p n, and G nd /G n+d 2 = p d 2 n. Continuing this process we obtain the nested sequence of subgroups G 0 G... G n... 8 and the sequence of elements g n G n \ G n+, such that G n+d ν /G n+d ν+ = p n, ν =, d. Theorem 3.. The topology and measure generated by the basic chain 8 coincide with the topology and measure generated by the chain G nd n = G d n n. Proof. Any coset G nd+l + h G nd is a finite union of cosets G n+d + j 0 g n, j g n,..., j d g n ; therefore the topologies generated by chains G n n and G nd n coincide. 632

7 S.F. Lukomskii Cosets G nd+l + h together with the empty set form a semiring M d. Define a measure µ on M d as follows: µg nd+l + h = µg nd+l =. p 0 p... p n d p l n Let µ be the corresponding outer measure. Cosets G nd + g together with the empty set form a semiring N d. Define a measure m on N d by mg nd + g = mg nd = p 0 p... p n. d Let m be the corresponding outer measure. It is enough to prove that m = µ. It is evident that N d M d, thus measures m and µ coincide on N d. Therefore, µ E m E for any E G. On the other hand, any coset G nd+l + h M d is a finite union of p d l n disjoint cosets G n+d + g N d and µg nd+l + h = p 0 p... p n d p l n = p 0 p... p n d pd l n = p d l n mg n+d + g = mg n+d + g. Therefore, any covering of the set E by cosets G nd+l + h M d is a covering by sets G n+d N d where the sum of measures of the covering cosets is preserved. Therefore, µ E m E and consequently µ E = m E. Choosing various basic chains and basic sequences we will obtain various Haar systems. It follows from [9] that all these Haar systems are complete orthonormal systems in L 2 G. Example 3.2. Let us construct subgroups G n+d ν and elements g n+d ν in the following way: and We can rewrite these identities in the form and In this case G n+d ν = p n Gn+d ν+ + jg n+d ν, g n+d ν = 0,..., 0, g }{{} n, 0,..., 0, ν =, d. d ν G nd+ν = p n Gnd+ν+ + jg nd+ν g nd+ν = 0,..., 0, g }{{} n, 0,..., 0, ν = 0, d. ν G nd+ν = G n+ G }{{ n+ G } n G n = G ν n+g d ν n. ν Since a character χ of the group G d is a product of characters of the group G, that is χx = χx 0, x,..., x d = χ 0 x 0 χ x... χ d x d, 9 χ belongs to the annulator Gnd+ν if and only if it can be represented in the form 9 with χ 0, χ,..., χ ν G n+ and χ ν,..., χ d G n ; i.e., Gnd+ν = G n+ G n+... G n+ G n... G n. }{{} ν 633

8 Haar system on a product of zero-dimensional compact groups Therefore, we can rewrite the Rademacher function r nd+ν G nd+ν+ \ G nd+ν in the form r nd+ν x = χ 0 x 0... χ ν x ν r n x ν χ ν+ x ν+... χ d x d, where χ 0,..., χ ν G n+ and χ ν+,..., χ d G n. Choosing characters χ j, j = 0,..., ν, ν +, ν + 2,..., d, equal to one, we obtain r nd+ν x = r n x ν, ν = 0, d, where r n x ν are one-dimensional Rademacher functions. If we substitute r nd+ν x as above to 6, then we get Haar functions in the form H 0 x, H lmn +kx = m N rnx l q GN +qx, where q = a 0 g 0 + a g + + a N g N, k = a 0 m 0 + a m + + a N m N, N = nd + ν, ν = 0, d. Taking into account definition of the elements g j, we get that q = q 0, q,..., q d, k = k 0 + k + + k d, where { a j g 0 + a d+j g + + a nd+j g n if j = 0, ν, q j = a j g 0 + a d+j g + + a n d+j g n if j = ν, d, { a j m d 0 k j = pj 0 + a d+jm d pj + + a nd+jm d np j n if j = 0, ν, a j m d 0 pj 0 + a d+jm d pj + + a n d+jm d n pj n if j = ν, d. By the equality r nd+ν x = r n x ν, we finally get H lmnd+ν +kx = m d np ν n r l nx ν q ν Gnd+ν +qx, where r n are one-dimensional Rademacher functions. If G = Z p, the Haar functions H lmnd+ν +kx may be obtained from the separable p-adic MRA see [2, 24]. Let us consider the case G = Z p, p = 2, d = 2, in detail. We have H x, x 2 = r 0 x G0 G 0 x, x 2, H 2 x, x 2 = 2 r 0 x 2 G G 0 x, x 2, H 3 x, x 2 = 2 r 0 x 2 G G 0 x g 0, x 2. Define a 2-dimensional dilation operator A 2 as follows. A 2 x, x 2 = x 2, Ax, where Ax = p x is the one-dimensional dilation operator. Then H 2 x, x 2 = 2 H A 2 x, x 2, H 3 x, x 2 = 2 H A2 x, x 2 g 0,

9 S.F. Lukomskii That is, we have unique wavelet function ψx, x 2 = r 0 x G0 G 0 x, x 2. Using the separable MRA [2] we will have three wavelet functions ψ {,2} x, x 2 = r 0 x r 0 x 2 G0 G 0 x, x 2 = 2 H 2 H 3 = H A 2 x, x 2 H A2 x, x 2 g 0, 0, ψ {} x, x 2 = r 0 x G0 G 0 x, x 2 = H x, x 2, ψ {2} x, x 2 = r 0 x 2 G0 G 0 x, x 2 = 2 H 2 + H 3 = H A 2 x, x 2 + H A2 x, x 2 g 0, 0, which can be written over unique wavelet function H. Here the dilation operator is à 2 x, x 2 = Ax, Ax 2. It is clear that Ã2 = A Double Haar system In this section, we describe all Haar systems on the product of two groups G = G G. Moreover, we will express them in terms of one-dimensional Rademacher functions. Any subgroup G 2n+ satisfying G 2n+ G 2n+ G 2n, is generated by an element αg n, βg n, where 0 α, β p n and α 2 + β 2. We can write this subgroup as G 2n+ = p n Gn+ G n+ + jαg n, βg n. If α = 0 or β = 0, we get G 2n+ = G n+ G n, or G 2n+ = G n G n+, respectively. Denote by G 2n+ a cyclic group, whose elements are cosets G n+ G n+ + jαg n, βg n, j = 0, p n. Lemma 4.. Let α, β. If G n+ + k 0 g n G n+ + k g n G 2n+, G n+ + l 0 g n G n+ + l g n G 2n+ are distinct cosets, then k 0 l 0 and k l. Proof. Let k 0 = l 0. If k 0 = l 0 = 0, then the set of cosets G n+ G n+ + kg n, k = 0, p n, is the group G 2n+. Consequently, G 2n+ = G n+ G n, which is impossible, since α 0. Let k 0 = l 0 0. In this case there exists j j such that G n+ + k 0 g n = G n+ + j αg n and G n+ + l 0 g n = G n+ + j αg n, which is impossible, since G n+ + jαg n pn is a cyclic group of prime order. 635

10 Haar system on a product of zero-dimensional compact groups Lemma 4.2. Any subgroup G 2n+ satisfying G 2n+ G 2n+ G 2n, can be represented as G 2n+ = p n Gn+ G n+ + jg n, νg n, ν = 0, p n, 0 or G 2n+ = p n Gn+ G n+ + j0, g n. Proof. By Lemma 4., any couple g n, νg n, ν = 0, p n, and 0, g n generate a group of cosets G n+ G n+ + jg n, νg n and G n+ G n+ + j0, g n, respectively. Thus, identities 0 and are straightforward. Corollary 4.3. The number of subgroups G 2n+ satisfying G 2n+ G 2n+ G 2n is equal to p n +. Theorem 4.4. Let g n, νg n generate the subgroup Let γ 0, γ, ξ 0, ξ = 0, p n, ξ 0, be integers such that Then, the functions G 2n+ = { G n+ G n+ + jg n, νg n } p n. γ 0 + γ ν 0 mod p n, 2 ξ 0 + ξ ν 0 mod p n. 3 r 2n+ x 0, x = r n γ 0 x 0 r n γ x, 4 r 2n x 0, x = r n ξ 0 x 0 r n ξ x, 5 where r n are one-dimensional Rademacher functions, are Rademacher functions on the group G = G G. Proof. Let us show that r 2n+ x 0, x G2n+ \ G 2n+ if γ 0 + γ ν 0 mod p n. Indeed, if x 0, x G 2n+, then we have r n γ 0 x 0 r n γ x = for any γ 0, γ. Let x 0, x G 2n+ \ G 2n+. Then there exists j 0, x n+, y n+ G n+ such that r 2n+ x 0, x = r n γ 0 x 0 r n γ x = r n γ0 x n+ + jg n r n γ y n+ + jνg n = r n jgn γ 0 + γ ν. Since γ 0 + γ ν 0 mod p n, r n jgn γ 0 + γ ν. Analogously, we can prove that r 2n G 2n+, when ξ 0 + ξ ν 0 mod p n ; i.e., r 2n G 2n+. Let us show that r 2nG 2n \ G 2n+. Indeed, if x 0, x G 2n \ G 2n+ then x 0, x G n+ G n+ + jg n, ν g n, j 0, ν ν 0. Therefore, r 2n x 0, x = r n jgn ξ 0 + ν ξ. Since ξ 0, ν ν, ξ 0 + ν ξ = αp n + ν νξ 0 mod p n, whence r n jgn ξ 0 + ν ξ. Thus r 2n G 2n \ G 2n+ and the theorem is proved. 636

11 S.F. Lukomskii Corollary 4.5. Since any character χ Gn+ \ G n can be represented in the form χ = rn αn r α n n rα 0 0, any Rademacher function r n x 0, x Gn+ \ G n can be represented as a product r n x 0, x = r αn n r α n n rα 0 0. Since the functions r n,..., r 0 are equal to on the subgroup G n, then we can rewrite Haar functions in the form H lmn+kx 0, x = r l nx q Gn +qx, where the functions r n are defined by 4 and 5. Therefore, equalities 4 and 5 together with r 2n+ x 0, x = r n x, r 2n x 0, x = r n x 0, define all Haar functions on the product G G. Example 4.6. Let us consider Haar functions H lmn +kx = r l Nx q GN +qx with N = 2n and N = 2n + in uniforormalization. Haar functions H lmn +kx are shifts of the function H lmn x. This function is constant on cosets G n+ G n+ + a 0 g n, a g n. Therefore, we can define the Haar function H lmn +kx with a matrix of dimension p n p n. For simplicity assume p n = 3. For G 2n+ we have the following possibilities: G 2n+ = 2 Gn+ G n+ + jg n, 0, G 2n+ = 2 Gn+ G n+ + jg n, g n, G 2n+ = 2 Gn+ G n+ + jg n, 2g n, G 2n+ = 2 Gn+ G n+ + j0, g n. Let us consider in detail the second case. Using Theorem 4.4, take r 2n x 0, x = r n x 0 r n 2x, r 2n+ x 0, x = r n x 0 r n x. For N = 2n there exist two Haar functions H mn x and H 2mN x that are nonzero on G n G n, and can be defined by the matrices ε ε 2 ε 2 ε ε 2 ε, ε ε 2 ε ε 2. ε 2 ε For N = 2n + we have six Haar functions H lmn +kx that are nonzero on G n G n. To define these Haar functions we need to take a basic element g 2n for which 2 G 2n = G2n+ + jg 2n. There are two possibilities: g 2n = g n, 2g n and g 2n = g n, 0. For g 2n = g n, 2g n we have three Haar functions H mn, H mn +m N = H mn g 2n, HmN +2m N = H mn 2g 2n 637

12 Haar system on a product of zero-dimensional compact groups with matrices 0 0 ε 2 0 ε 0, ε 2 0 0, 0 0 ε ε , 0 ε 2 0 and three Haar functions H 2mN, H 2mN +m N, H 2mN +2m N with matrices 0 0 ε 0 ε 2 0, ε 0 0, 0 0 ε 2 ε ε 0 For g 2n = g n, 0 we have three Haar functions H mn, HmN +m N, HmN +2m N with matrices 0 0 ε 2 0 ε 0, 0 0 ε ε, ε 2 0 ε 0 0, 0 0 and three Haar functions H 2mN, H2mN +m N, H2mN +2m N with matrices 0 0 ε 0 ε 2 0, 0 0 ε ε 2, ε 0 ε Other Haar functions H lmn +k can be obtained by shifts of vectors h = h 0, h, where h 0 = a 0 g 0 + a g + + a n g n, h = b 0 g 0 + b g + + b n g n. Acknowledgements This research was carried out with the financial support of the Programme for Support of Leading Scientific Schools of the President of the Russian Federation grant no. NSh and the Russian Foundation for Basic Research grant no References [] Agaev G.N., Vilenkin N.Ya., Dzhafarli G.M., Rubinshteĭn A.I., Multiplicative Systems and Harmonic Analysis on Zero-Dimensional Groups, ELM, Baku, 98 in Russian [2] Albeverio S., Khrennikov A.Yu., Shelkovich V.M., Theory of p-adic Distributions: Linear and Nonlinear Models, London Math. Soc. Lecture Note Ser., 370, Cambridge University Press, Cambridge, 200 [3] Albeverio S., Evdokimov S., Skopina M., p-adic nonorthogonal wavelet bases, Tr. Mat. Inst. Steklova, 2009, 265, Izbrannye Voprosy Matematicheskoi Fiziki i p-adicheskogo Analiza, 7 8 [4] Albeverio S., Evdokimov S., Skopina M., p-adic multiresolution analysis and wavelet frames, J. Fourier Anal. Appl., 200, 65, [5] Benedetto J.J., Benedetto R.L., A wavelet theory for local fields and related groups, J. Geom. Anal., 2004, 43, [6] Benedetto R.L., Examples of wavelets for local fields, In: Wavelets, Frames and Operator Theory, Contemp. Math., 345, American Mathematical Society, Providence, 2004,

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