We study the connection between STFT multipliers $A^{\f_1,\f_2}_{1\otimes m}$ having windows $\f_1,\f_2$, symbols $a\phas=(1\otimes m)\phas=m(\o)$, $\phas\in\rdd$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\rd$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $\f_1,\f_2$ for the equality $T_{m_2}= A^{\f_1,\f_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{\f_1,\f_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called \emph{two-window short-time Fourier transform} which enters in the definition of $A^{\f_1,\f_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{\f,\f}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.

Comparisons between Fourier and STFT multipliers: The smoothing effect of the short-time Fourier transform

Bastianoni F.;Cordero E.
;
Feichtinger H. G.;
In corso di stampa

Abstract

We study the connection between STFT multipliers $A^{\f_1,\f_2}_{1\otimes m}$ having windows $\f_1,\f_2$, symbols $a\phas=(1\otimes m)\phas=m(\o)$, $\phas\in\rdd$, and the Fourier multipliers $T_{m_2}$ with symbol $m_2$ on $\rd$. We find sufficient and necessary conditions on symbols $m,m_2$ and windows $\f_1,\f_2$ for the equality $T_{m_2}= A^{\f_1,\f_2}_{1\otimes m}$. For $m=m_2$ the former equality holds only for particular choices of window functions in modulation spaces, whereas it never occurs in the realm of Lebesgue spaces. In general, the STFT multiplier $A^{\f_1,\f_2}_{1\otimes m}$, also called localization operator, presents a smoothing effect due to the so-called \emph{two-window short-time Fourier transform} which enters in the definition of $A^{\f_1,\f_2}_{1\otimes m}$. As a by-product we prove necessary conditions for the continuity of anti-Wick operators $A^{\f,\f}_{1\otimes m}: L^p\to L^q$ having multiplier $m$ in weak $L^r$ spaces. Finally, we exhibit the related results for their discrete counterpart: in this setting STFT multipliers are called Gabor multipliers whereas Fourier multiplier are better known as linear time invariant (LTI) filters.
In corso di stampa
529
1
1
32
https://www.sciencedirect.com/science/article/pii/S0022247X23005826
Localization operators; Modulation spaces; Short-time Fourier transform; STFT multipliers; Time-frequency analysis; Wiener amalgam spaces
Balazs P.; Bastianoni F.; Cordero E.; Feichtinger H.G.; Schweighofer N.
File in questo prodotto:
File Dimensione Formato  
2024JMAA.pdf

Accesso riservato

Tipo di file: PDF EDITORIALE
Dimensione 1.13 MB
Formato Adobe PDF
1.13 MB Adobe PDF   Visualizza/Apri   Richiedi una copia

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/1928499
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 0
  • ???jsp.display-item.citation.isi??? 0
social impact