An Operator Based Approach to Irregular Frames of Translates

We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Peter Balazs [verfasserIn] Sigrid Heineken [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2019

Schlagwörter:	frames Riesz bases irregular translates canonical duals frame-related operators

Übergeordnetes Werk:	In: Mathematics - MDPI AG, 2013, 7(2019), 5, p 449
Übergeordnetes Werk:	volume:7 ; year:2019 ; number:5, p 449

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/math7050449

Katalog-ID:	DOAJ079622836

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callnumber-first	Q - Science
author	Peter Balazs
spellingShingle	Peter Balazs misc QA1-939 misc frames misc Riesz bases misc irregular translates misc canonical duals misc frame-related operators misc Mathematics An Operator Based Approach to Irregular Frames of Translates
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topic_title	QA1-939 An Operator Based Approach to Irregular Frames of Translates frames Riesz bases irregular translates canonical duals frame-related operators
topic	misc QA1-939 misc frames misc Riesz bases misc irregular translates misc canonical duals misc frame-related operators misc Mathematics
topic_unstemmed	misc QA1-939 misc frames misc Riesz bases misc irregular translates misc canonical duals misc frame-related operators misc Mathematics
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title	An Operator Based Approach to Irregular Frames of Translates
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title_full	An Operator Based Approach to Irregular Frames of Translates
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title_sort	operator based approach to irregular frames of translates
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title_auth	An Operator Based Approach to Irregular Frames of Translates
abstract	We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.
abstractGer	We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.
abstract_unstemmed	We consider translates of functions in <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mi<L</mi< <mn<2</mn< </msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi mathvariant="double-struck"<R</mi< <mi<d</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< along an irregular set of points, that is, <inline-formula< <math display="inline"< <semantics< <msub< <mrow< <mo stretchy="false"<{</mo< <mi<ϕ</mi< <mrow< <mo stretchy="false"<(</mo< <mo<·</mo< <mo<−</mo< <msub< <mi<λ</mi< <mi<k</mi< </msub< <mo stretchy="false"<)</mo< </mrow< <mo stretchy="false"<}</mo< </mrow< <mrow< <mi<k</mi< <mo<∈</mo< <mi mathvariant="double-struck"<Z</mi< </mrow< </msub< </semantics< </math< </inline-formula<—where <inline-formula< <math display="inline"< <semantics< <mi<ϕ</mi< </semantics< </math< </inline-formula< is a bandlimited function. Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. This is used, in particular, to find for the first time in the irregular case a representation of the canonical dual as well as of the equivalent Parseval frame—in terms of its Fourier transform.
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container_issue	5, p 449
title_short	An Operator Based Approach to Irregular Frames of Translates
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Introducing a notion of pseudo-Gramian function for the irregular case, we obtain conditions for a family of irregular translates to be a Bessel, frame or Riesz sequence. We show the connection of the frame-related operators of the translates to the operators of exponentials. 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An Operator Based Approach to Irregular Frames of Translates

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