An Operator Based Approach to Irregular Frames of Translates

Peter Balazs and Sigrid Heineken
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Peter Balazs: Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, Austria
Sigrid Heineken: IMAS UBA-CONICET, Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabellón I, Ciudad Universitaria, C1428EGA Buenos Aires, Argentina

Mathematics, 2019, vol. 7, issue 5, 1-11

Abstract: We consider translates of functions in L 2 ( R d ) along an irregular set of points, that is, { ? ( · − λ k ) } k ∈ Z —where ? 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.

Keywords: frames; Riesz bases; irregular translates; canonical duals; frame-related operators (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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