An Operator Based Approach to Irregular Frames of Translates
Acoustics Research Institute, Austrian Academy of Sciences, Wohllebengasse 12-14, 1040 Wien, Austria
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.
Received: 29 March 2019 / Revised: 12 May 2019 / Accepted: 13 May 2019 / Published: 20 May 2019
PDF [278 KB, uploaded 20 May 2019]
We consider translates of functions in
along an irregular set of points, that is,
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.
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).
Share & Cite This Article
MDPI and ACS Style
Balazs, P.; Heineken, S. An Operator Based Approach to Irregular Frames of Translates. Mathematics 2019, 7, 449.
Balazs P, Heineken S. An Operator Based Approach to Irregular Frames of Translates. Mathematics. 2019; 7(5):449.
Balazs, Peter; Heineken, Sigrid. 2019. "An Operator Based Approach to Irregular Frames of Translates." Mathematics 7, no. 5: 449.
Show more citation formats
Show less citations formats
Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.
[Return to top]
For more information on the journal statistics, click here
Multiple requests from the same IP address are counted as one view.