Time and band limiting for matrix valued functions: an integral and a commuting differential operator. (English) Zbl 1367.47078
Summary: The problem of recovering a signal of finite duration from a piece of its Fourier transform was solved at Bell Labs in the 1960’s, by exploiting a ‘miracle’: a certain naturally appearing integral operator commutes with an explicit differential one. Here we show that this same miracle holds in a matrix valued version of the same problem.
MSC:
| 47N99 | Miscellaneous applications of operator theory |
| 94A12 | Signal theory (characterization, reconstruction, filtering, etc.) |
| 33C47 | Other special orthogonal polynomials and functions |
| 47E05 | General theory of ordinary differential operators |
| 47G10 | Integral operators |
References:
| [1] | Castro M and Grünbaum F A 2015 The Darboux process and time-and-band limiting for matrix orthogonal polynomials Linear Algebr. Appl.487 328–41 |
| [2] | Davison M E 1983 The ill-conditioned nature of the limited angle tomography problem SIAM J. Appl. Math.43 428–48 · Zbl 0526.44005 |
| [3] | Davison M E and Grünbaum F A 1979 Convolution algorithms for arbitrary projections angles IEEE Trans. Nucl. Sci.26 2670–3 |
| [4] | Davison M E and Grünbaum F A 1981 Tomographic reconstruction with arbitrary directions Commun. Pure Appl. Math.34 77–119 · Zbl 0441.93031 |
| [5] | Durán A J and Grünbaum F A 2004 Orthogonal matrix polynomials satisfying second-order differential equations Int. Math. Res. Not.10 461–84 |
| [6] | Durán A J 1996 Markov’s theorem for orthogonal matrix polynomials Can. J. Math.48 1180–95 |
| [7] | Grünbaum F A, Longhi L and Perlstadt M 1982 Differential operators commuting with finite convolution integral operators: some nonabelian examples SIAM J. Appl. Math.42 941–55 · Zbl 0497.22012 |
| [8] | Grünbaum F A, Pacharoni I and Tirao J 2003 Matrix valued orthogonal polynomials of the Jacobi type Indagationes Math.14 353–66 · Zbl 1070.33011 |
| [9] | Grünbaum F A, Pacharoni I and Zurrián I 2015 Time and band limiting for matrix valued functions, an example SIGMA Symmetry Integrability Geom. Methods Appl.11 14 |
| [10] | Grünbaum F A 1986 Some mathematical problems motivated by medical imaging Inverse Problems (Montecatini Terme, 1986)((Lecture Notes in Mathematics vol 1225)) (Berlin: Springer) pp 113–41 · Zbl 0648.42004 |
| [11] | Gullberg G T, Roy D G, Zeng G L, Alexander A L and Parker D L 1999 Tensor tomography IEEE Trans. Nucl. Sci.46 991–1000 |
| [12] | Katsnelson V 2016 Self-adjoint boundary conditions for the prolate spheroid differential operator (arXiv:1603.07542) |
| [13] | Katsnelson V and Machluf R 2012 The truncated Fourier operator, general results Zh. Mat. Fiz. Anal. Geom.8 158–76 · Zbl 1286.47022 |
| [14] | Louis A K 1986 Incomplete data problems in x-ray computerized tomography I. Singular value decomposition of the limited angle transform Numer. Math.48 251–62 |
| [15] | Landau H J and Pollak H O 1961 Prolate spheroidal wave functions, Fourier analysis and uncertainty II Bell Syst. Tech. J.40 65–84 · Zbl 0184.08602 |
| [16] | Landau H J and Pollak H O 1962 Prolate spheroidal wave functions, Fourier analysis and uncertainty III. The dimension of the space of essentially time- and band-limited signals Bell Syst. Tech. J.41 1295–336 · Zbl 0184.08603 |
| [17] | Landau H J and Pollak H O 1978 Prolate spheroidal wave functions, Fourier analysis and uncertainty V Bell Syst. Tech. J.57 1371–430 |
| [18] | Natterer F 2001 The Mathematics of Computerized Tomography (Philadelphia, PA: SIAM) · Zbl 0973.92020 |
| [19] | Osipov A, Rokhlin V and Xiao H 2013 Prolate Spheroidal Wave Functions of Order Zero: Mathematical Tools for Bandlimited Approximation(Applied Mathematical Sciences vol 187) (New York: Springer) · Zbl 1287.65015 |
| [20] | Paternain G P, Salo M and Uhlmann G 1999 Tensor tomography: progress and challenges Chin. Ann. Math. Ser. B 35 399–428 · Zbl 1303.92053 |
| [21] | Pacharoni I and Zurrián I 2016 Matrix gegenbauer polynomials: the 2{\(\times\)}2 fundamental cases. Constr. Approx.43 253–71 · Zbl 1342.33025 |
| [22] | Simons F J and Dahlen F A 2006 Spherical slepian functions on the polar gap in geodesy Geophys. J. Int.166 1039–61 |
| [23] | Simons F J, Dahlen F A and Wieczorek M A 2006 Spatiospectral concentration on a sphere SIAM Rev.48 504–36 |
| [24] | Slepian D 1964 Prolate spheroidal wave functions, Fourier analysis and uncertainity IV. Extensions to many dimensions; generalized prolate spheroidal functions Bell Syst. Tech. J.43 3009–57 · Zbl 0184.08604 |
| [25] | Slepian D and Pollak H O 1961 Prolate spheroidal wave functions, Fourier analysis and uncertainty I. Bell Syst. Tech. J.40 43–63 · Zbl 0184.08601 |
| [26] | Tirao J and Zurrián I 2014 Spherical functions of fundamental K-types associated with the n-dimensional sphere. SIGMA, Symmetry Integrability Geom. Methods Appl.10 41 · Zbl 1297.22011 |
| [27] | Zurrián I 2016 The algebra of differential operators for a gegenbauer weight matrix Int. Math. Res. Not.2016 1–29 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
