# Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets

Studia Mathematica (1996)

• Volume: 119, Issue: 1, page 37-64
• ISSN: 0039-3223

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## Abstract

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We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g↦{P}_{g,\varphi }$, where for a fixed function ϕ, ${P}_{g,\varphi }$ denotes the one-dimensional orthogonal projection on the function ${U}_{g}\varphi$, U is a group representation and g is an element of the group. They are defined as integrals ${ʃ}_{W}{P}_{g,\varphi }dg$, where W is an open, relatively compact subset of a group. Our main result is a characterization of function spaces corresponding to local Toeplitz operators with pth power summable eigenvalues, 0 < p ≤ ∞.

## How to cite

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Nowak, Krzysztof. "Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets." Studia Mathematica 119.1 (1996): 37-64. <http://eudml.org/doc/216285>.

@article{Nowak1996,
abstract = {We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g ↦ P_\{g,ϕ\}$, where for a fixed function ϕ, $P_\{g,ϕ\}$ denotes the one-dimensional orthogonal projection on the function $U_gϕ$, U is a group representation and g is an element of the group. They are defined as integrals $ʃ_W P_\{g,ϕ\} dg$, where W is an open, relatively compact subset of a group. Our main result is a characterization of function spaces corresponding to local Toeplitz operators with pth power summable eigenvalues, 0 < p ≤ ∞.},
author = {Nowak, Krzysztof},
journal = {Studia Mathematica},
keywords = {singular values; time-frequency localization; rough wavelets; group representations; dilations; translations; modulations; local Toeplitz operator; left invariant measure; th power summable eigenvalues; Schatten ideal; integral lattice; Fourier transform; inverse Fourier transform},
language = {eng},
number = {1},
pages = {37-64},
title = {Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets},
url = {http://eudml.org/doc/216285},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Nowak, Krzysztof
TI - Local Toeplitz operators based on wavelets: phase space patterns for rough wavelets
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 1
SP - 37
EP - 64
AB - We consider two standard group representations: one acting on functions by translations and dilations, the other by translations and modulations, and we study local Toeplitz operators based on them. Local Toeplitz operators are the averages of projection-valued functions $g ↦ P_{g,ϕ}$, where for a fixed function ϕ, $P_{g,ϕ}$ denotes the one-dimensional orthogonal projection on the function $U_gϕ$, U is a group representation and g is an element of the group. They are defined as integrals $ʃ_W P_{g,ϕ} dg$, where W is an open, relatively compact subset of a group. Our main result is a characterization of function spaces corresponding to local Toeplitz operators with pth power summable eigenvalues, 0 < p ≤ ∞.
LA - eng
KW - singular values; time-frequency localization; rough wavelets; group representations; dilations; translations; modulations; local Toeplitz operator; left invariant measure; th power summable eigenvalues; Schatten ideal; integral lattice; Fourier transform; inverse Fourier transform
UR - http://eudml.org/doc/216285
ER -

## References

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