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Open AccessArticle

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)
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
MDPI and ACS Style

Balazs, P.; Heineken, S. An Operator Based Approach to Irregular Frames of Translates. Mathematics 2019, 7, 449.

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