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An Operator Based Approach to Irregular Frames of Translates

1
Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, Austria
2
IMAS UBA-CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA Buenos Aires, Argentina
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(5), 449; https://doi.org/10.3390/math7050449
Received: 29 March 2019 / Revised: 12 May 2019 / Accepted: 13 May 2019 / Published: 20 May 2019
(This article belongs to the Special Issue Harmonic Analysis)
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PDF [278 KB, uploaded 20 May 2019]

Abstract

We consider translates of functions in L 2 ( R d ) along an irregular set of points, that is, { ϕ ( · λ k ) } k Z —where ϕ 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. View Full-Text
Keywords: frames; Riesz bases; irregular translates; canonical duals; frame-related operators frames; Riesz bases; irregular translates; canonical duals; frame-related operators
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Balazs, P.; Heineken, S. An Operator Based Approach to Irregular Frames of Translates. Mathematics 2019, 7, 449.

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