φ-FRAMES AND φ-RIESZ BASES ON LOCALLY COMPACT ABELIAN GROUPS

This paper develops several aspects of shift-invariant spaces on locally compact abelian groups. For a second countable locally compact abelian group G we prove a useful Hilbert space isomorphism, introduce range functions and give a characterization of shift-invariant subspaces of L2(G) in terms of range functions. Utilizing these functions, we generalize characterizations of frames and Riesz bases generated by shifts of a countable set of generators from L2(ℝn) to L2(G).

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Scaling sets and generalized scaling sets on Cantor dyadic group

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691320500198 ◽

2020 ◽

Vol 18 (04) ◽

pp. 2050019

Author(s):

Prasadini Mahapatra ◽

Divya Singh

Keyword(s):

Abelian Groups ◽

Real Case ◽

Locally Compact ◽

Wavelet Sets ◽

Locally Compact Abelian Groups ◽

Dyadic Group ◽

Compact Abelian Groups ◽

Cantor Dyadic Group

Scaling and generalized scaling sets determine wavelet sets and hence wavelets. In real case, wavelet sets were proved to be an important tool for the construction of MRA as well as non-MRA wavelets. However, any result related to scaling/generalized scaling sets is not available in case of locally compact abelian groups. This paper gives a characterization of scaling sets and its generalized version along with relevant examples in dual Cantor dyadic group [Formula: see text]. These results can further be generalized to arbitrary locally compact abelian groups.

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Semisimple Banach Algebras Generated by Strongly Continuous Representations of Locally Compact Abelian Groups

Journal of Functional Analysis ◽

10.1006/jfan.1994.1138 ◽

1994 ◽

Vol 126 (1) ◽

pp. 1-6 ◽

Cited By ~ 3

Author(s):

G. Muraz ◽

V.Q. Phong

Keyword(s):

Abelian Groups ◽

Banach Algebras ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups ◽

Continuous Representations

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Singular measures with absolutely continuous convolution squares

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100039992 ◽

1966 ◽

Vol 62 (3) ◽

pp. 399-420 ◽

Cited By ~ 38

Author(s):

Edwin Hewitt ◽

Herbert S. Zuckerman

Keyword(s):

Abelian Groups ◽

Natural Order ◽

Continuous Measure ◽

Locally Compact ◽

Absolutely Continuous ◽

Character Group ◽

Locally Compact Abelian Groups ◽

Singular Measures ◽

Compact Abelian Groups ◽

Nikodym Derivative

Introduction. A famous construction of Wiener and Wintner ((13)), later refined by Salem ((11)) and extended by Schaeffer ((12)) and Ivašev-Musatov ((8)), produces a non-negative, singular, continuous measure μ on [ − π,π[ such thatfor every ∈ > 0. It is plain that the convolution μ * μ is absolutely continuous and in fact has Lebesgue–Radon–Nikodým derivative f such that For general locally compact Abelian groups, no exact analogue of (1 · 1) seems possible, as the character group may admit no natural order. However, it makes good sense to ask if μ* μ is absolutely continuous and has pth power integrable derivative. We will construct continuous singular measures μ on all non-discrete locally compact Abelian groups G such that μ * μ is a absolutely continuous and for which the Lebesgue–Radon–Nikodým derivative of μ * μ is in, for all real p > 1.

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