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Rashidi-Kouchi M. Wavelet Sets on Locally Compact Abelian Groups. mmr 2020; 6 (3) :393-404

URL: http://mmr.khu.ac.ir/article-1-2657-en.html

URL: http://mmr.khu.ac.ir/article-1-2657-en.html

An orthonormal wavelet is a square-integrable function whose translates and dilates form an orthonormal basis for the Hilbert space . That is, given the unitary operators of translation

for and dilation , we call an orthonormal wavelet if the set

is an orthonormal basis for . This definition was later generalized to higher dimensions and to allow for other dilation and translation sets; let Hilbert space and an

is an orthogonal basis for .

The concept of a multiresolution analysis, abbreviated as MRA is Central to the theory of wavelets. There is much overlaps between wavelet analysis and Fourier analysis. Indeed, wavelets can be thought of as non-trigonometric Fourier series. Thus, Fourier analysis is used as a tool to investigate properties of wavelets.

Another concept is wavelet set. The term wavelet set was coined by Dai and Larson in

the late 90s to describe a set

In this paper, we investigate wavelet sets on locally compact abelian groups with uniform lattice, where a uniform lattice H in LCA group G is a discrete subgroup of G such that the quotient group G/H is compact. So we review some basic facts from the theory of LCA groups and harmonic analysis. Then we define wavelet sets on these groups and characterize them by using of Fourier transform and multiresolution analysis.

We extend theory of wavelet sets on locally compact abelian groups with uniform lattice. This is a generalization of wavelet sets on Euclidean space. We characterize wavelet

sets by using of Fourier transform and multiresolution analysis. Also, we define generalized scaling sets and dimension functions on locally compact abelian groups and verify its relations with wavelet sets. Dimension functions for MSF wavelets are described by generalized scaling sets.

In the setting of LCA groups, we define translation congruent and show wavelet sets are translation congruent, so we can define a map on G such that it is measurable, measure preserving and bijection.

The following conclusions were drawn from this research.

- Wavelet sets on locally compact groups by uniform lattice can be defined. This is a generalization of wavelet sets on Euclidean space.
- Characterization of wavelet sets on LCA groups can be done in different ways. A method is to use Fourier transform and translation congruent. Another way is to generaliz scaling set and dimension function.
- As an example, Cantor dyadic group is a non-trivial example that satisfies in the theory of wavelet sets on locally compact groups by uniform lattice. We find wavelet set and generalized scaling set for this group and show related wavelet is MRA wavelet../files/site1/files/63/7.pdf

Type of Study: Research Paper |
Subject:
alg

Received: 2017/07/23 | Revised: 2021/01/4 | Accepted: 2019/01/14 | Published: 2020/11/30 | ePublished: 2020/11/30

Received: 2017/07/23 | Revised: 2021/01/4 | Accepted: 2019/01/14 | Published: 2020/11/30 | ePublished: 2020/11/30

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