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An Operator Based Approach to Irregular Frames of Translates

Author

Listed:
  • 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)

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.

Suggested Citation

  • Peter Balazs & Sigrid Heineken, 2019. "An Operator Based Approach to Irregular Frames of Translates," Mathematics, MDPI, vol. 7(5), pages 1-11, May.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:5:p:449-:d:232698
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