Duffin and Schaeffer [10] introduced the notion of frames, and Daubechies et al. [9] the notion of wavelet frames. Among frames, tight wavelet frames, due to their numerical stability, play an essential role in the series representation of a function and signal transmission. Other types of wavelets with simple structure have been established on a Lebesgue measurable set in Rⁿ with a non-negative and finite measure. The most fundamental one is the Haar wavelet Ψ(x)=1[0,1/2]−1[1/2,1], introduced by Alfred Haar in 1910. It is a one-dimensional prototypical illustration of a wavelet. The Shannon wavelet, whose Fourier transform is 1[−1,−1/2)∪(1/2,1], is another typical example. The Shannon wavelet is an essential tool for analyzing and reconstructing functions. We concentrate on Shannon-type wavelets whose Fourier transform is supported in the finite measure domain. We study Shannon wavelet's Fourier transforms in the Shannon set Θ=[−2π,−π]∪[π,2π). We take the orthonormal wavelet Ψ=2sinh(2x−1)−sinh(x) with Ψ^=1Θ, where Ψ^ is a Fourier transform of Ψ and 1Θ is an indicator of set Θ. For more examples and constructions of wavelet sets in Rⁿ see [2, 3, 7, 23, 24]. Wang [29] addressed the existence of wavelet sets with the documentation of spectral sets. The characterization of wavelets has been generalized to n-dimensional Euclidean spaces. The dilation set is defined on any countable subset of non-singular matrices, whereas the translation set is defined on any countable subset in Rⁿ.

Grepstad and Lev [14, 15] introduced the concept of multi-tiling and Riesz bases on a bounded Riemann measurable set in Rⁿ. Shah et al. [25] provided complete characterizations of orthogonal families, tight frames, and orthonormal bases of Gabor systems on local fields of positive characteristic. More results in this direction can be found in [5, 6, 26, 28]. Iosevich et al. [19] gave the concept of tight wavelet frame sets in finite vector spaces. This paper continues the line of research for locally compact abelian (LCA) groups. Mayeli [21] gave the concept of Riesz wavelets, tiling, and spectral sets in LCA groups. Further motivated by works of [1, 4, 12, 20, 27], we introduce the sufficient condition for the existence of tight wavelet frame sets in LCA groups. We also construct an orthonormal wavelet basis for L²(G).

Some LCA groups, such as the rational p-adic additive group Q {p}, have no discrete subgroup in the field of p-adic rational numbers. Consequently, such groups do not apply to the classical concepts of wavelet sets. Thus the finite abelian group does not possess any wavelet set in the conventional sense. Iosevich et al. [19] defined tilings of tight wavelet frame sets in finite vector space. We introduce the concept of tilings of tight wavelet frame sets in infinite LCA groups. We assume that G is an infinite group and admits a discrete subgroup. For the notion of wavelet sets in Q_p and a non-commutative setting we refer [3] and ]8], respectively. For Riesz wavelets' characterizations generated by multiresolution analysis in the finite cases, see [4, 16, 22].

The main objective of this work is to investigate the characterization of tight wavelet frame sets for infinite LCA groups equipped with a Haar measure. We present two approaches to construct (for stationary and non-stationary wavelets) the scaling function for L²(G). We construct an orthonormal wavelet basis for L²(G). We propose an open problem related to the extension principle for Riesz wavelets in locally compact abelian groups.

Now we give some well-known definitions and results.

Definition 2.1.([19])

A subset {xi}i∈Z of a Hilbert space H is said to be a frame if there exists a sequence of vectors {xi:i∈Z} in H and two positive real numbers P≤Q such that for every x∈H

P and Q are the lower and upper bounds of the frame. If P = Q , then the frame is called a tight wavelet frame and if P = Q = 1 , then the frame is called a Parseval frame.

Definition 2.2.([21]) A collection of Haar-measurable wavelet sets {Θl}l=1r in Rⁿ is called a Riesz wavelet set if there exists a countable set S⊂GL(n,R) and Ψl∈Rn such that for Ψ^l=1Θl, where GL(n,R) is a general linear group of order n. The collection of elements

is called a Riesz wavelet basis for L2(Rn), where Δ is a unitary operator over R and r is a fixed positive integer.

The main feature of frames is their redundancy; for example, they play an essential role in robustness. In Hilbert's space, the frame behaves as a simplified, visualized characterization of sets. Engineers and applied mathematicians have used frames for frequency-domain analysis of discrete-time systems. Frames are more attractive in the space L²(G) as it is closed to the mother wavelet.

Let α,β,γ∈R be given. We define the linear operators: modulation Mα, dilation Dβ and translation Tγ, for a function f∈L2(G) as follows:

If {Mα,Dβ,Tγ} is a frame in L2(G), then (α,β,γ) generates a wavelet frame in L2(G). The Fourier transform of a function f∈L2(G) is defined by f^(ω)=∫G f(x)e−2πiωxdx. The Fourier transform of modulation Mα, dilation Dβ and translation Tγ, is defined by Mαf^=M−αf^, Dβf^=D1/βf^ and Tγf^=Tγf^ respectively.

Definition 2.3.([21]) Let Ψ∈L2(Rn), n ≥ 1, then Ψ is called a wavelet for L2(Rn) if there exist D ⊂GL(n,R) and countable sets T⊂Rn, such that the family

is an orthonormal basis for L2(Rn). The sets D and T are called dilations and translation sets, respectively.

The size of D and T of orthonormal wavelet frames for L2(Rn) have been studied by many authors, see [17, 18, 29].

Definition 2.4.([21]) If G is an LCA group equipped with a Haar measure and G^ its dual. A collection of measurable subsets Θk, 1≤k≤n, of G^ is

called a (Riesz) wavelet collection of sets for G if there is a countable subset A of Aut(G) and countable subsets Tk⊂G (1≤k≤n) such that

is a (Riesz) orthogonal basis for L²(G), Ψk,j,r(x)=△(r)1/2Ψk(r(x)−λ), Ψ^k=1Θk When n = 1, we may say Θ=Θ1 is a (Riesz) wavelet set.

Wavelet sets and minimally supported frequency wavelets were introduced by Fang and Wang [11] and studied exclusively by Hernandez et al. [17, 18]. A measurable set Θ⊂Rn with non-zero finite measure is called a wavelet set, if for the function Ψ∈L2(Rn) with Ψ^=1Θ and the set ω of (2.2) becomes an orthogonal basis for L2(Rn). A simple way to establish a wavelet frame in Rⁿ is to choose a function whose Fourier transform is the indicator of a measurable set. The classical interpretation of wavelet frame sets obtains dilations and translations of the family (2.4). A set Θ⊂Rn is called a wavelet frame set with respect to dilations and translations if for the function Ψ∈L2(Rn) with Ψ^=1Θ, the family (2.4) becomes a frame in L2(Rn).

Wang [29] studied the characterization of wavelet sets. Shah and Debnath [26] developed tight wavelet frames using the unitary extension principle in local fields and gave sufficient conditions for the formation of tight wavelet frame sets with a limited number of refinable functions. Wavelet sets were discussed by many authors [5, 13]. Mayeli [21] gave the concept of tilings of Riesz wavelets in LCA groups. Let G be an infinite LCA group equipped with Haar measure and C a set of complex numbers. A linear function Ψ:G→C is said to be a wavelet frame for L²(G) if there exists a subset A⊂Aut(G) and a countable subset T □ G such that the family

is an orthonormal basis with respect to wavelet frames for L²(G). Note that in continuous cases, for a non-singular matrix M, the dilation factor c=|detM|n/2 makes the dilation map f→|det(M)|1/2f(Mx) an isometry. If c=1 then the dilation of function in Rⁿ belongs to a discrete group. To overcome this problem in the discrete case, we take both stationary and non-stationary wavelet settings of LCA groups to define the class of automorphism in G. In this way, the dilation of a function in Rⁿ can be obtained in a co-compact group as well as a discrete group. In [2, 19, 26] the relationship between spectral sets and translational tilings were discussed. A measurable set Θ with positive measure is called a spectral set if there exists a countable set T such that the collection of exponentials {e2πiδx:δ∈T} forms an orthonormal basis for L2(Θ). In this case, the pair (Θ,T) is said to be a spectral pair.

Proposition 2.5.([29]) Let D ∈ GL(n,R), T⊂Rn and Θ⊂Rn be a positive finite Lebesgue measure. If {Mt(Θ):M∈D} is a tiling of Rⁿ and (Θ,T) is a spectral pair, then Ψ=1ˇΘ is a wavelet with respect to the dilation set D and the translation set T. Conversely, if Ψ=1ˇΘ is a wavelet with respect to the dilation set D and the translation set T and 0 ∈ T, then {Mt(Θ):M∈D} is a tiling of Rⁿ and (Θ,T) is a spectral pair.

Definition 2.6.([21]) Let Θ be a subset of Rⁿ, then Θ is said to be a multiplicative tiling set for Rⁿ corresponding to a collection of n×n non-singular matrices M if {α(Θ):α∈M} is a set partition for Rⁿ i.e.,

Equivalently, Rn=∪ α∈Mα(Θ) where for any α≠α' the two sets α(Θ) and α'(Θ) are disjoint in measure.

Definition 2.7.([21]) Let Θ be a subset of Rⁿ, then Θ is said to be a translational tiling set for Rⁿ corresponding a countable set L⊂Rn if

Equivalently, Rn=∪ l∈LΘ+l, where for each pair of independent elements l and l', the sets Θ +l and Θ+l' disjoint in measure.

Let f be a function on an infinite LCA group G; then the Fourier coefficient is defined by

where χn(ξ)=e2πin.ξα is the characteristic function, α is in G, n.ξ is an inner product and we have a_n=f(n). Here n is a variable. The function f^ is the Fourier transform of f. The Fourier transform of the function f ∈ G is defined as

where bn=f^(n). The Fourier transform F:L2(G)→L2(G) given by f→f^ is a unitary map. For any h∈L2(G), we shall denote by hˇ the inverse Fourier transform of h.

Let Aut(G) be the set of all automorphism of G. A linear function Ψ:L2(G)→C is a wavelet frame if there exists a set of automorphism A⊂Aut(G) and a subset T⊂G such that the linear map

is an orthonormal basis concerning a wavelet frame for L2(G).

When α∈Aut(G) and β ∈ G is associated with dilation and translation operators σα and τ_β defined by

respectively.

Definition 3.1. A measurable subset Θ⊂Rn with finite and positive measure is called a wavelet set if there is a function Ψ∈L2(Rn) with Ψ^=1Θ, such that the family 2.4 is an orthonormal basis for L²(G) .

Definition 3.2. Let J be a subset of an infinite LCA group G, then J is called a multiplicative tiling for G if there exists a set of group automorphism A in Aut(G) such that J tiles G multiplicatively by A, i.e.,

for α≠α' the union of two sets α(J) and α'(J) are disjoint.

Definition 3.3. Let J be a subset of an infinite LCA group G, then J is called a translational tiling set for G if there exists a subset T⊆G such that

and for β≠β' the union of the two sets (J+β) and (J+β') is disjoint.

As a result, if J is a multiplicative tiling set for an infinite LCA group G and 0 ∈ G, then α(0)=0 is a natural requirement for any group automorphism α. We say a set J has spectrum K if the characteristic {χk}k∈K is an orthonormal basis for L²(J). In this case, we say J is a spectral and (J, K) is a spectral pair.

Proposition 3.4. Let G be an infinite LCA group, C a set of complex numbers and Ψ:L2(G)→C, then for a certain group automorphism α∈Aut(G) and β∈G we have

where α*=(αt)−1=(α−1)t the inverse transpose of α.

If Ψ^=1J, where J is a spectral set, then

Proof. Let α∈Aut(G) and β ∈ G. By applying a Fourier transform and modifying the variable using the definition of Fourier transforms, for all n ∈ G we have,

and

Now a combination of both equations (3.1) and (3.2) gives the assertion of the first part of Proposition 3.4.:

For the proof of the second part of proposition 3.4., we take the equality 1J(α*n)=1αt(J)(n), then the proposition also holds.

{\bf Proposition 3.5.} Let A⊆Aut(G) and T⊆G. The family

is an orthonormal basis for L²(G) if and only if the family

is an orthonormal basis for L²(G). Here n is a variable.

The proof of Proposition 3.5. is straightforward using a Fourier transform and Proposition 3.4.

Propostion 3.6. There exists no non empty subset J⊊G such that Ψ=1ˇJ, the inverse Fourier transform of the indicator function 1_J, is a generator of Parseval wavelet frame for L²(G).

Proof. Now we have a contradictory argument to prove this proposition.

Suppose that there exists a non-empty set J and a subset A⊂Aut(G) and a set T⊂G such that the set

is a Parseval frame for L²(G). By Proposition 3.5., we get that

is a Parseval frame for L2(G)=L2(G^), where G^ is a dual group of an infinite LCA group G. Let h^=1{0}∈L2(G) be a indicator function for the set {0}. Then

where # stands for number of elements. Two cases are taken into consideration here:

Case one: If 0 ∈ J, then 0∈αt(J) for all α ∈ A. The above formula thus indicates that 1=♯(A)♯(T). So there must be only one element of the wavelet system, let A={α} and T={β}. Then all the vectors in L²(G) must be constant multiple of σατβΨ. We show that this is not the case if J ≠ G. Let g≠ 0 in L²(G) such that (g^)∩αt(J) is empty. There is no such function since J is not the whole group G. Then there is no constant ϵ such that g=ϵσατβΨ. This indicates that there is not just one element to the Parseval wavelet frame when J≠G, thus ♯(A)♯(T)>1.

Case two: If 0 does not belong to J. By the equation in (3.4) we get 1=0, which is impossible. This completes the proof.

Therefore, the above Proposition 3.6. shows that there does not exist any Parseval wavelet frame for L²(G) generated by Ψ=1ˇJ.

Definition 3.7. Let J and K be two subsets of an LCA group G, and we say (J, K) is a tight wavelet frame spectral pair if the set of characteristic {χk:k∈K} is a tight wavelet frame for L²(G).

Proposition 3.8. Let J and K be two subsets of G and (J, K) be a spectral pair. Suppose that J is a multiplicative tiling set concerning a set of automorphisms A⊂Aut(G). The following hold true:

• (i) (J,K) is a tight wavelet frame spectral pair with the frame bound 0<P≤Q<∞.

• (ii) For all α ∈ A, (α(J),(α−1)t(K)) is a tight wavelet frame spectral pair with the frame bound 0<P≤Q<∞.

• (iii) The family {1α(J)χ(α−1)t(k):α∈A,k∈K} is a tight wavelet frame for L²(G) with the frame bound 0<P≤Q<∞.

• (iv) If 0 not in J, then {(J)−1/21α(J)χ(α−1)t(k):α∈A,k∈K} is an orthonormal basis for L²(G).

Proof. Suppose J has a spectrum K. Then {(J)−1/2χk:k∈K} is an orthonormal basis for L²(G).

Proof of (i): Note that a projection map f→f1J of L²(G) onto L²(G), therefore the image of the orthonormal basis {(J)−1/2χk:k∈K} by this projection map is a Parseval frame for L²(G). This shows the statement (i).

Proof of (ii): Let H be a subgroup of LCA group G and {χk:k∈K} be a frame for L²(H) with the frame bound 0<P≤Q<∞. Then {χ(α−1)t(k):k∈K} is a frame for L2(α(H)) with the unified frame bounds P and Q. To prove this, we define the map Fα:L2(H)→L2(α(H)) by f→f∘α−1. The image of {χk:k∈K} under F_α is {1α(H)χ(α−1)t(k):k∈K} and this map is unitary. Therefore, {1α(H)χ(α−1)t(k):k∈K} forms a frame for L²(α(H)) with the same frame bounds.

Proof of (iii): By the supposition on the multiplicative tiling property of J we get

To complete (iii), we shall prove the following instead.

Let Y be a measurable set and I be an index set such that Y=∪ α∈AYα are disjoint. Suppose that for all α ∈ A, L2(Yα) has a tight wavelet frame {fm,α:m∈Iα} with the frame bound S. We are saying that the family {fm,α:α∈A,m∈Iα} is a tight wavelet frame for L²(Y) with the frame bound S. To show this, let h∈L2(Y). Since {Yα:α∈A} is a partition for Y, then h=⊕α∈Ahα, hα=h1Yα and we get

This completes the proof of (iii).

Proof of (iv): If J is a spectral set in with 0 does not belong to J; therefore, facts can be derived explicitly from the fact that any Parseval frame with normalized frame components is an orthonormal basis.

Now, we propose an open problem about the extension principle for Riesz wavelets in LCA groups.

Let M be a non-singular matrix with dilation factor c=|detM|1/2 and L a subset of an LCA group G, then for j ≥ 0, we define the 2πM−jl shift operator

Our aim is to construct scaling functions Φjr r=1,2,⋯m in L²(G) and for the Riesz wavelets Ψjr, r=1,2,⋯ρj in L²(G), where m and ρ_j are positive integers, such that the collection

forms a Riesz wavelet orthonormal basis for L²(G).

In the way defined above, what type of scaling function can show the extension principle for Riesz wavelets in LCA groups?