Spatiospectral concentration in the Cartesian plane

Frederik Jozef Simons, Dong V. Wang

Research output: Contribution to journalArticle

26 Scopus citations

Abstract

We pose and solve the analogue of Slepian's time-frequency concentration problem in the two-dimensional plane, for applications in the natural sciences. We determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the plane, or, alternatively, of strictly spacelimited functions that are optimally concentrated in the Fourier domain. The Cartesian Slepian functions can be found by solving a Fredholm integral equation whose associated eigenvalues are a measure of the spatiospectral concentration. Both the spatial and spectral regions of concentration can, in principle, have arbitrary geometry. However, for practical applications of signal representation or spectral analysis such as exist in geophysics or astronomy, in physical space irregular shapes, and in spectral space symmetric domains will usually be preferred. When the concentration domains are circularly symmetric in both spaces, the Slepian functions are also eigenfunctions of a Sturm-Liouville operator, leading to special algorithms for this case, as is well known. Much like their one-dimensional and spherical counterparts with which we discuss them in a common framework, a basis of functions that are simultaneously spatially and spectrally localized on arbitrary Cartesian domains will be of great utility in many scientific disciplines, but especially in the geosciences.

Original languageEnglish (US)
Pages (from-to)1-36
Number of pages36
JournalGEM - International Journal on Geomathematics
Volume2
Issue number1
DOIs
StatePublished - Jun 2011

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Earth and Planetary Sciences(all)

Keywords

  • Bandlimited function
  • Commuting differential operator
  • Concentration problem
  • Eigenvalue problem
  • Reproducing kernel
  • Spectral analysis
  • Spherical harmonics
  • Sturm-Liouville problem

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