GABOR MULTIPLIERS FOR BANACH SPACES OF DISTRIBUTIONS ON LOCALLY COMPACT ABELIAN GROUPS

2006 ◽  
Author(s):  
S. S. PANDEY
Keyword(s):  
Banach Spaces ◽  
Abelian Groups ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

scholarly journals Uniform Boundedness for Groups

1962 ◽  
Vol 14 ◽  
pp. 269-276 ◽  
Author(s):  
Irving Glicksberg
Keyword(s):  
Banach Spaces ◽  
Present Note ◽  
Abelian Groups ◽  
Uniform Boundedness ◽  
Striking Similarity ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

Let G and H be locally compact abelian groups with character groups G*, H*, and let < . , . > denote the pairing between a group and its dual.In 1952 Kaplansky proved the following result, using the structure of locally compact abelian groups and category arguments.Theorem 1.1. Let τ: G → H be an algebraic homomorphism for which there is a dual τ* : H* → G* (so that < rg, h* > = < g, τ*h* > for all g in G, h* in H*). Then τ is continuous.The result bears a striking similarity to a well-known fact about Banach spaces which is a consequence of uniform boundedness; the present note is devoted to an analogous “uniform boundedness” for groups, which yields a non-structural proof of Kaplansky's theorem.

Scaling sets and generalized scaling sets on Cantor dyadic group

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.

Singular measures with absolutely continuous convolution squares

1966 ◽  
Vol 62 (3) ◽  
pp. 399-420 ◽  
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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