ON FUNCTION SPACES CHARACTERIZED BY THE WIGNER TRANSFORM

Year 2021, Volume: 4 Issue: 2, 188 - 200, 31.07.2021

Öznur Kulak

https://doi.org/10.33773/jum.958029

Cited By: 1

Abstract

Let $\omega _{i}$ be weight
functions on $%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, (i=1,2,3,4). In this work, we define $CW_{\omega _{1},\omega _{2},\omega _{3},\omega
_{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ to be vector space of $\left( f,g\right) \in \left( L_{\omega
_{1}}^{p}\times L_{\omega _{2}}^{q}\right) \left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ such that the $\tau -$Wigner transforms $W_{\tau }\left(
f,.\right) $ and $W_{\tau }\left( .,g\right) $ belong to $L_{\omega
_{3}}^{r}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}\right) $ and $L_{\omega _{4}}^{s}\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}\right) $ respectively for $1\leq p,q,r,s<\infty $, $\tau \in \left(
0,1\right) $. We endow this space with a sum norm and prove that $%
CW_{\omega _{1},\omega _{2},\omega _{3},\omega _{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ is a Banach space. We also show that $CW_{\omega _{1},\omega
_{2},\omega _{3},\omega _{4}}^{p,q,r,s,\tau }\left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $ becomes an essential Banach module over $\left( L_{\omega
_{1}}^{1}\times L_{\omega _{2}}^{1}\right) \left(
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
\right) $. We then consider approximate identities.

Keywords

Wigner transform, Essential Banach module, Approximate identity

Supporting Institution

Giresun University

Project Number

FEN-BAP-C-150219-01

References

• P. Boggiatto, G. De Donno, A. Oliaro, A class of quadratic time- frequency representations based on the short- time Fourier transform, Oper Theory, 172, 235-249, (2007).
• P. Boggiatto, G. De Donno, A. Oliaro, Time- frequency representations of Wigner type and pseudo- differential operators, Trans Amer Math Soc, 362, 4955-4981, (2010).
• R.S. Doran, J. Wichmann, Approximate identity and factorization in Banach modules, Lecture Notes in Math. Springer-Verlag, 768 (1979).
• M. Duman, Ö. Kulak, On Function Spaces with Fractional Wavelet Transform, Montes Taurus J. Pure Appl. Math. 3 (3), 122–134 (2021).
• R.H. Fischer, A.T. Gürkanlı, T.S. Liu, On a family of weighted spaces, Mathematica Slovaca, 46(1), 71-82 (1996).
• I.G. Gaudry, Multipliers of weighted Lebesgue and measure spaces, Proc.Lon.Math.Soc., 19(3), 327-340 (1969).
• K. Gröchenig, Foundations of Time-Frequency Analysis, Birkhauser, Boston (2001).
• Ö. Kulak, A.T. Gürkanlı, On Function Spaces with Wavelet Transform in L-omega-p-R, Hacettepe Journal of Mathematics and Statistics, 40(2), 163-177 (2011).
• H. Reiter, Classical Harmonic Analysis and Locally Compact Group, Oxford Universty Pres, Oxford (1968).
• A. Sandıkçı, Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces, Turkish Journal of Mathematics, 38, 728- 745 (2014).
• A. Sandıkçı, Multilinear tau -wigner transform, J. Pseudo-Differ. Oper. Appl., 11, 1465-1487 (2020).
• H.C. Wang, Homogeneous Banach algebras, New York: Marcel Dekker Inc. (1977).