Fourier Frames for Surface-Carried Measures

Abstract In this paper, we show that the surface measure on the boundary of a convex body of everywhere positive Gaussian curvature does not admit a Fourier frame. This answers a question proposed by Lev and provides the 1st example of a uniformly distributed measure supported on a set of Lebesgue measure zero that does not admit a Fourier frame. In contrast, we show that the surface measure on the boundary of a polytope always admits a Fourier frame. We also explore orthogonal bases and frames adopted to sets under consideration. More precisely, given a compact manifold $M$ without a boundary and $D \subset M$, we ask whether $L^2(D)$ possesses an orthogonal basis of eigenfunctions. The non-abelian nature of this problem, in general, puts it outside the realm of the previously explored questions about the existence of bases of characters for subsets of locally compact abelian groups. This paper is dedicated to Alexander Olevskii on the occasion of his birthday. Olevskii’s mathematical depth and personal kindness serve as a major source of inspiration for us and many others in the field of mathematics.

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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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On the Frames of Translates on Locally Compact Abelian Groups

Bulletin of the Iranian Mathematical Society ◽

10.1007/s41980-021-00643-2 ◽

2021 ◽

Author(s):

Najmeh Sadat Seyedi ◽

Rajab Ali Kamyabi Gol

Keyword(s):

Abelian Groups ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups

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Embeddings of locally compact abelian p-groups in Hawaiian groups

Forum Mathematicum ◽

10.1515/forum-2021-0085 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Yanga Bavuma ◽

Francesco G. Russo

Keyword(s):

Algebraic Topology ◽

Geometric Interpretation ◽

Locally Compact Groups ◽

Compact Groups ◽

Locally Compact ◽

Locally Compact Abelian Groups ◽

Compact Abelian Groups ◽

Hawaiian Earring ◽

Path Connected

Abstract We show that locally compact abelian p-groups can be embedded in the first Hawaiian group on a compact path connected subspace of the Euclidean space of dimension four. This result gives a new geometric interpretation for the classification of locally compact abelian groups which are rich in commuting closed subgroups. It is then possible to introduce the idea of an algebraic topology for topologically modular locally compact groups via the geometry of the Hawaiian earring. Among other things, we find applications for locally compact groups which are just noncompact.

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