Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 11 (2015), 044, 14 pages      arXiv:1410.1232      https://doi.org/10.3842/SIGMA.2015.044
Contribution to the Special Issue on Exact Solvability and Symmetry Avatars in honour of Luc Vinet

Time and Band Limiting for Matrix Valued Functions, an Example

F. Alberto Grünbaum a, Inés Pacharoni b and Ignacio Nahuel Zurrián b
a) Department of Mathematics, University of California, Berkeley 94705, USA
b) CIEM-FaMAF, Universidad Nacional de Córdoba, Córdoba 5000, Argentina

Received February 11, 2015, in final form May 30, 2015; Published online June 12, 2015

Abstract
The main purpose of this paper is to extend to a situation involving matrix valued orthogonal polynomials and spherical functions, a result that traces its origin and its importance to work of Claude Shannon in laying the mathematical foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of ''time and band limiting'' admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.

Key words: time-band limiting; double concentration; matrix valued orthogonal polynomials.

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References

  1. Bonami A., Karoui A., Uniform approximation and explicit estimates for the prolate spheroidal wave functions, arXiv:1405.3676.
  2. Demmel J.W., Applied numerical linear algebra, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1997.
  3. Duistermaat J.J., Grünbaum F.A., Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240.
  4. Grünbaum F.A., A new property of reproducing kernels for classical orthogonal polynomials, J. Math. Anal. Appl. 95 (1983), 491-500.
  5. Grünbaum F.A., Some new explorations into the mystery of time and band limiting, Adv. in Appl. Math. 13 (1992), 328-349.
  6. Grünbaum F.A., Band-time-band limiting integral operators and commuting differential operators, St. Petersburg Math. J. 8 (1997), 93-96.
  7. Grünbaum F.A., The bispectral problem: an overview, in Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., Vol. 30, Kluwer Acad. Publ., Dordrecht, 2001, 129-140.
  8. Grünbaum F.A., Longhi L., Perlstadt M., Differential operators commuting with finite convolution integral operators: some nonabelian examples, SIAM J. Appl. Math. 42 (1982), 941-955.
  9. Grünbaum F.A., Yakimov M., The prolate spheroidal phenomenon as a consequence of bispectrality, in Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, Vol. 37, Amer. Math. Soc., Providence, RI, 2004, 301-312, math-ph/0303041.
  10. Jahn K., Bokor N., Revisiting the concentration problem of vector fields within a spherical cap: a commuting differential operator solution, J. Fourier Anal. Appl. 20 (2014), 421-451, arXiv:1302.5261.
  11. Jamming P., Karoui A., Spektor S., The approximation of almost time and band limited functions by their expansion in some orthogonal polynomial bases, arXiv:1501.03655.
  12. Landau H.J., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. II, Bell System Tech. J. 40 (1961), 65-84.
  13. Landau H.J., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals, Bell System Tech. J. 41 (1962), 1295-1336.
  14. Melkman A.A., $n$-widths and optimal interpolation of time- and band-limited functions, in Optimal Estimation in Approximation Theory (Proc. Internat. Sympos., Freudenstadt, 1976), Editors C.A. Michelli, T. Rivlin, Plenum, New York, 1977, 55-68.
  15. Osipov A., Rokhlin V., Xiao H., Prolate spheroidal wave functions of order zero. Mathematical tools for bandlimited approximation, Applied Mathematical Sciences, Vol. 187, Springer, New York, 2013.
  16. Pacharoni I., Zurrian I., Matrix ultraspherical polynomials: the $2\times 2$ fundamental cases, Constr. Approx., to appear, arXiv:1309.6902.
  17. Parlett B.N., The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980.
  18. Perlstadt M., Chopped orthogonal polynomial expansions - some discrete cases, SIAM J. Algebraic Discrete Methods 4 (1983), 94-100.
  19. Perlstadt M., A property of orthogonal polynomial families with polynomial duals, SIAM J. Math. Anal. 15 (1984), 1043-1054.
  20. Plattner A., Simons F.J., Spatiospectral concentration of vector fields on a sphere, Appl. Comput. Harmon. Anal. 36 (2014), 1-22, arXiv:1306.3201.
  21. Shannon C.E., A mathematical theory of communication, Bell System Tech. J. 27 (1948), 379-423.
  22. Shannon C.E., A mathematical theory of communication, Bell System Tech. J. 27 (1948), 623-656.
  23. Simons F.J., Dahlen F.A., Spherical Slepian functions on the polar gap in geodesy, Geophys. J. Int. 166 (2006), 1039-1061, math.ST/0603271.
  24. Simons F.J., Dahlen F.A., Wieczorek M.A., Spatiospectral concentration on a sphere, SIAM Rev. 48 (2006), 504-536, math.CA/0408424.
  25. Slepian D., Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions, Bell System Tech. J. 43 (1964), 3009-3057.
  26. Slepian D., On bandwidth, Proc. IEEE 64 (1976), 292-300.
  27. Slepian D., Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. The discrete case, Bell System Tech. J. 57 (1978), 1371-1430.
  28. Slepian D., Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev. 25 (1983), 379-393.
  29. Slepian D., Pollak H.O., Prolate spheroidal wave functions, Fourier analysis and uncertainty. I, Bell System Tech. J. 40 (1961), 43-63.
  30. Stewart G.W., Matrix algorithms. Vol. II. Eigensystems, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2001.
  31. Tirao J.A., Zurrián I.N., Spherical functions of fundamental $K$-types associated with the $n$-dimensional sphere, SIGMA 10 (2014), 071, 41 pages, arXiv:1312.0909.
  32. Weyl H., The theory of groups and quantum mechanics, Dutton, New York, 1931.

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