B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

Ole Christensen, Hong Oh Kim, Rae Young Kim

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.
Original languageEnglish
JournalJournal of Fourier Analysis and Applications
Volume24
Issue number4
Pages (from-to)1119-1140
ISSN1069-5869
DOIs
Publication statusPublished - 2017

Keywords

  • B-Splines
  • Dual frames
  • Frames
  • Gaussian

Cite this

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title = "B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames",
abstract = "We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to {\^a}€œalmost perfect reconstruction{\^a}€� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.",
keywords = "B-Splines, Dual frames, Frames, Gaussian",
author = "Ole Christensen and Kim, {Hong Oh} and Kim, {Rae Young}",
year = "2017",
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language = "English",
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journal = "Journal of Fourier Analysis and Applications",
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B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames. / Christensen, Ole; Kim, Hong Oh; Kim, Rae Young.

In: Journal of Fourier Analysis and Applications, Vol. 24, No. 4, 2017, p. 1119-1140.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - B-Spline Approximations of the Gaussian, their Gabor Frame Properties, and Approximately Dual Frames

AU - Christensen, Ole

AU - Kim, Hong Oh

AU - Kim, Rae Young

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N2 - We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.

AB - We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters (Formula presented.) with (Formula presented.), the scaled version of (Formula presented.) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction� within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the (Formula presented.)-spaces, as well in the time-domain as in the frequency domain.

KW - B-Splines

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