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Realizable algorithm for approximating Hilbert-Schmidt operators via Gabor Multipliers.

Darian M Onchis1,2, Simone Zappalà1,2

  • 1Darian M. Onchis is with the Faculty of Mathematics, University of Vienna, Austria and Faculty of Mathematics and Computer Science, West University of Timisoara, Romania.

Journal of Computational and Applied Mathematics
|March 10, 2018
PubMed
Summary
This summary is machine-generated.

This study enhances the approximation of Hilbert-Schmidt operators using generalized Gabor multipliers. New methods improve accuracy, especially for complex systems with discontinuities, by integrating feature detection with harmonic analysis.

Keywords:
Gabor multipliersHilbert-Schmidt operators approximationHough transformcomputational algorithmspline-type spaces

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Area of Science:

  • Harmonic Analysis
  • Numerical Analysis
  • Signal Processing

Background:

  • Hilbert-Schmidt operators are crucial in various scientific fields.
  • Generalized Gabor multipliers offer a framework for operator approximation.
  • Existing methods face challenges with discontinuous systems.

Purpose of the Study:

  • To introduce novel computational techniques for approximating Hilbert-Schmidt operators.
  • To enhance the accuracy of generalized Gabor multipliers, particularly for overspread operators.
  • To address the challenge of approximating discontinuities in operator symbols.

Main Methods:

  • Approximating operator symbols via L2 projection in spline-type spaces.
  • Utilizing a selection procedure for analysis and synthesis functions (time-frequency lag).
  • Employing Hough transform for discontinuity detection and smoothing with new approximants.

Main Results:

  • The proposed algorithm effectively handles both underspread and overspread operators.
  • Significant improvements in approximating overspread operators with smooth spreading functions.
  • The Hough transform integration nearly doubles approximation accuracy for discontinuous systems.

Conclusions:

  • The integration of feature detection (Hough transform) with harmonic analysis substantially improves operator approximation.
  • The developed methods offer a robust approach for handling complex, time-varying systems.
  • This work bridges signal processing and harmonic analysis for enhanced computational efficiency.