On a certain class of positive definite functions and measures on locally compact Abelian groups and inner-product spaces

2022 ◽  
Vol 282 (1) ◽  
pp. 109261
Author(s):  
Wojciech Banaszczyk
Keyword(s):  
Abelian Groups ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Inner Product ◽  
Product Spaces ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Inner Product Spaces ◽  
Compact Abelian Groups

scholarly journals Resonance classes of measures

1987 ◽  
Vol 10 (3) ◽  
pp. 461-471 ◽  
Author(s):  
Maria Torres De Squire
Keyword(s):  
Fourier Analysis ◽  
Real Line ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Locally Compact ◽  
The Real ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups ◽  
Definition Of ◽  
Amalgam Spaces

We extendF. Holland's definition of the space of resonant classes of functions, on the real line, to the spaceR(Φpq) (1≦p, q≦∞)of resonant classes of measures, on locally compact abelian groups. We characterize this space in terms of transformable measures and establish a realatlonship betweenR(Φpq)and the set of positive definite functions for amalgam spaces. As a consequence we answer the conjecture posed by L. Argabright and J. Gil de Lamadrid in their work on Fourier analysis of unbounded measures.

Positive Definite Measures with Discrete Fourier Transform and Pure Point Diffraction

2011 ◽  
Vol 54 (3) ◽  
pp. 544-555 ◽  
Author(s):  
Nicolae Strungaru
Keyword(s):  
Fourier Transform ◽  
Discrete Fourier Transform ◽  
Abelian Groups ◽  
Positive Definite ◽  
Pure Point ◽  
Locally Compact ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractIn this paper we characterize the positive definite measures with discrete Fourier transform. As an application we provide a characterization of pure point diffraction in locally compact Abelian groups.

Positive Definiteness on Products of Compact Two-point Homogeneous Spaces and Locally Compact Abelian Groups

2019 ◽  
Vol 63 (4) ◽  
pp. 705-715
Author(s):  
V. A. Menegatto ◽  
C. P. Oliveira
Keyword(s):  
Homogeneous Spaces ◽  
Abelian Groups ◽  
Geodesic Distance ◽  
Positive Definiteness ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Locally Compact ◽  
Profile Function ◽  
Locally Compact Abelian Groups ◽  
Compact Abelian Groups

AbstractIn this paper, we consider the problem of characterizing positive definite functions on compact two-point homogeneous spaces cross locally compact abelian groups. For a locally compact abelian group $G$ with dual group $\widehat{G}$, a compact two-point homogeneous space $\mathbb{H}$ with normalized geodesic distance $\unicode[STIX]{x1D6FF}$ and a profile function $\unicode[STIX]{x1D719}:[-1,1]\times G\rightarrow \mathbb{C}$ satisfying certain continuity and integrability assumptions, we show that the positive definiteness of the kernel $((x,u),(y,v))\in (\mathbb{H}\times G)^{2}\mapsto \unicode[STIX]{x1D719}(\cos \unicode[STIX]{x1D6FF}(x,y),uv^{-1})$ is equivalent to the positive definiteness of the Fourier transformed kernels $(x,y)\in \mathbb{H}^{2}\mapsto \widehat{\unicode[STIX]{x1D719}}_{\cos \unicode[STIX]{x1D6FF}(x,y)}(\unicode[STIX]{x1D6FE})$, $\unicode[STIX]{x1D6FE}\in \widehat{G}$, where $\unicode[STIX]{x1D719}_{t}(u)=\unicode[STIX]{x1D719}(t,u)$, $u\in G$. We also provide some results on the strict positive definiteness of the kernel.

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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