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Time and Band Limiting for Matrix Valued Functions, an Example
Symmetry, integrability and geometry: methods and applications, Tome 11 (2015) Cet article a éte moissonné depuis la source Math-Net.Ru

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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.
Keywords: time-band limiting; double concentration; matrix valued orthogonal polynomials. 
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F. Alberto Grünbaum; Inés Pacharoni; Ignacio Nahuel Zurrián. Time and Band Limiting for Matrix Valued Functions, an Example. Symmetry, integrability and geometry: methods and applications, Tome 11 (2015). http://geodesic.mathdoc.fr/item/SIGMA_2015_11_a43/

[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, SIAM, Philadelphia, PA, 1997 | DOI | MR | Zbl

[3] Duistermaat J. J., Grünbaum F. A., “Differential equations in the spectral parameter”, Comm. Math. Phys., 103 (1986), 177–240 | DOI | MR | Zbl

[4] Grünbaum F. A., “A new property of reproducing kernels for classical orthogonal polynomials”, J. Math. Anal. Appl., 95 (1983), 491–500 | DOI | MR | Zbl

[5] Grünbaum F. A., “Some new explorations into the mystery of time and band limiting”, Adv. in Appl. Math., 13 (1992), 328–349 | DOI | MR | Zbl

[6] Grünbaum F. A., “Band-time-band limiting integral operators and commuting differential operators”, St. Petersburg Math. J., 8 (1997), 93–96 | MR

[7] Grünbaum F. A., “The bispectral problem: an overview”, Special Functions 2000: Current Perspective and Future Directions (Tempe, AZ), NATO Sci. Ser. II Math. Phys. Chem., 30, Kluwer Acad. Publ., Dordrecht, 2001, 129–140 | DOI | MR | Zbl

[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 | DOI | MR

[9] Grünbaum F. A., Yakimov M., “The prolate spheroidal phenomenon as a consequence of bispectrality”, Superintegrability in Classical and Quantum Systems, CRM Proc. Lecture Notes, 37, Amer. Math. Soc., Providence, RI, 2004, 301–312, arXiv: math-ph/0303041 | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[14] Melkman A. A., “$n$-widths and optimal interpolation of time- and band-limited functions”, Optimal Estimation in Approximation Theory, Proc. Internat. Sympos. (Freudenstadt, 1976), eds. C. A. Michelli, T. Rivlin, Plenum, New York, 1977, 55–68 | DOI | MR

[15] Osipov A., Rokhlin V., Xiao H., Prolate spheroidal wave functions of order zero. Mathematical tools for bandlimited approximation, Applied Mathematical Sciences, 187, Springer, New York, 2013 | DOI | MR | Zbl

[16] Pacharoni I., Zurrian I., “Matrix ultraspherical polynomials: the $2\times 2$ fundamental cases”, Constr. Approx. (to appear)

[17] Parlett B. N., The symmetric eigenvalue problem, Prentice-Hall Series in Computational Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1980 | MR | Zbl

[18] Perlstadt M., “Chopped orthogonal polynomial expansions — some discrete cases”, SIAM J. Algebraic Discrete Methods, 4 (1983), 94–100 | DOI | MR | Zbl

[19] Perlstadt M., “A property of orthogonal polynomial families with polynomial duals”, SIAM J. Math. Anal., 15 (1984), 1043–1054 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[21] Shannon C. E., “A mathematical theory of communication”, Bell System Tech. J., 27 (1948), 379–423 | DOI | MR | Zbl

[22] Shannon C. E., “A mathematical theory of communication”, Bell System Tech. J., 27 (1948), 623–656 | DOI | MR | Zbl

[23] Simons F. J., Dahlen F. A., “Spherical Slepian functions on the polar gap in geodesy”, Geophys. J. Int., 166 (2006), 1039–1061, arXiv: math.ST/0603271 | DOI

[24] Simons F. J., Dahlen F. A., Wieczorek M. A., “Spatiospectral concentration on a sphere”, SIAM Rev., 48 (2006), 504–536, arXiv: math.CA/0408424 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[26] Slepian D., “On bandwidth”, Proc. IEEE, 64 (1976), 292–300 | DOI | MR

[27] Slepian D., “Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV: The discrete case”, Bell System Tech. J., 57 (1978), 1371–1430 | DOI | Zbl

[28] Slepian D., “Some comments on Fourier analysis, uncertainty and modeling”, SIAM Rev., 25 (1983), 379–393 | DOI | MR | Zbl

[29] Slepian D., Pollak H. O., “Prolate spheroidal wave functions, Fourier analysis and uncertainty, I”, Bell System Tech. J., 40 (1961), 43–63 | DOI | MR | Zbl

[30] Stewart G. W., Matrix algorithms, v. II, Eigensystems, SIAM, Philadelphia, PA, 2001 | DOI | MR | Zbl

[31] Tirao J. A., Zurrián I. N., “Spherical functions of fundamental $K$-types associated with the $n$-dimensional sphere”, SIGMA, 10 (2014), 071, 41 pp., arXiv: 1312.0909 | DOI | MR | Zbl

[32] Weyl H., The theory of groups and quantum mechanics, Dutton, New York, 1931 | Zbl