Introduction Matrix Orthogonal Polynomials and band and time limiting Matrix Orthogonal Polynomials and Time and Band limiting Mirta M. Castro Smirnova Universidad de Sevilla, Espa˜ na Joint work with F. Alberto Gr¨ unbaum, University of California, Berkeley, I. Pacharoni, CIEM-FaMAF, Universidad Nacional de C´ ordoba, Argentina, I. Zurri´ an, Pontificia Universidad Cat´ olica, Santiago, Chile OPCOP, 19-22 de abril 2017, Castro Urdiales, Cantabria Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Outline Introduction 1 The origins of time and band limiting Setting of the problem in the matrix case Matrix Orthogonal Polynomials and band and time limiting 2 The discrete version of the problem The continuous version of the problem, an example Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Outline Introduction 1 The origins of time and band limiting Setting of the problem in the matrix case Matrix Orthogonal Polynomials and band and time limiting 2 The discrete version of the problem The continuous version of the problem, an example Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting C. Shannon, A mathematical theory of communication , Bell Tech. J., vol 27 , 1948, pp 379–423 (July) and pp 623–656 (Oct). Shannon’s Problem: Consider an unknow signal f ( t ) of finite duration, i.e. with support in [ − T, T ] (time limiting); the data are the values of its Fourier transform F ( k ) for k in the interval [ −W , W ] . (band limiting) In practice one only has noisy values of F ( k ) . What is the best use of this information? Though this problem was posed originally by Shannon, a full solution can be found in joint works by: David Slepian, Henry Landau and Henry Pollak. (Bell labs,1960’s) Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting C. Shannon, A mathematical theory of communication , Bell Tech. J., vol 27 , 1948, pp 379–423 (July) and pp 623–656 (Oct). Shannon’s Problem: Consider an unknow signal f ( t ) of finite duration, i.e. with support in [ − T, T ] (time limiting); the data are the values of its Fourier transform F ( k ) for k in the interval [ −W , W ] . (band limiting) In practice one only has noisy values of F ( k ) . What is the best use of this information? Though this problem was posed originally by Shannon, a full solution can be found in joint works by: David Slepian, Henry Landau and Henry Pollak. (Bell labs,1960’s) Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting C. Shannon, A mathematical theory of communication , Bell Tech. J., vol 27 , 1948, pp 379–423 (July) and pp 623–656 (Oct). Shannon’s Problem: Consider an unknow signal f ( t ) of finite duration, i.e. with support in [ − T, T ] (time limiting); the data are the values of its Fourier transform F ( k ) for k in the interval [ −W , W ] . (band limiting) In practice one only has noisy values of F ( k ) . What is the best use of this information? Though this problem was posed originally by Shannon, a full solution can be found in joint works by: David Slepian, Henry Landau and Henry Pollak. (Bell labs,1960’s) Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting In certain areas of Mathematics one arrives at a global operator, given by an integral kernel or a full matrix. One needs to compute numerically many of its eigenfunctions in an efficient way. In certain cases one can exhibit a differential operator of low order, or a narrow band matrix, which has the same eigenfuntions as the global operator. The numerical computation of the eigenfunctions of the differential operator, a local one, is much easier than the initial task. Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting Shannon- Slepian- Landau- Pollak : A = [ − T, T ] , B = [ −W , W ] , � T sin( W ( x − y )) ( I f )( x ) = f ( y ) dy, x ∈ A. x − y − T The operator ( Df )( x ) = (( T 2 − x 2 ) f ′ ( x )) ′ − W 2 x 2 f has simple spectrum and an appropriate selfadjoint extension of D commutes with I . Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting The kernel is obtained by integrating e isx in [ −W , W ] and one has an integral operator acting on functions defined in [ − T, T ] . � W k ( x, y ) = sin( W ( x − y )) e isx e − isy ds = x − y −W Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting How to find (and explain the existence of...) the differential operator D that will commute with an integral operator I with kernel k ( x, y ) acting on L 2 [ a, b ] ? Some useful guide-references F. A. Gr¨ unbaum, Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions , SIAM J. on Algebraic Discrete Methods 2 (1981), 136–141. F. A. Gr¨ unbaum, A new property of reproducing kernels of classical orthogonal polynomials , J. Math. Anal. Applic. 95 (1983), 491–500. F. A. Gr¨ unbaum, L. Longhi, and M. Perlstadt, Differential operators commuting with finite convolution integral operators: some nonabelian examples , SIAM J. Appl. Math. 42 (1982), 941–955. Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting The origins of time and band limiting The origins of time and band limiting How to find (and explain the existence of...) the differential operator D that will commute with an integral operator I with kernel k ( x, y ) acting on L 2 [ a, b ] ? Some useful guide-references F. A. Gr¨ unbaum, Eigenvectors of a Toeplitz matrix: discrete version of the prolate spheroidal wave functions , SIAM J. on Algebraic Discrete Methods 2 (1981), 136–141. F. A. Gr¨ unbaum, A new property of reproducing kernels of classical orthogonal polynomials , J. Math. Anal. Applic. 95 (1983), 491–500. F. A. Gr¨ unbaum, L. Longhi, and M. Perlstadt, Differential operators commuting with finite convolution integral operators: some nonabelian examples , SIAM J. Appl. Math. 42 (1982), 941–955. Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Setting of the problem in the matrix case The matrix case In this talk we will consider an example of matrix valued orthogonal polynomials satisfying differential equations (i.e a bispectral situation) in connection with time and band limiting. Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Setting of the problem in the matrix case “discrete version” of time and band limiting: one deals with a global operator given by a full matrix and one looks for a commuting local object given by a tridiagonal matrix. “continuous version” of time and band limiting: In this talk we deal with a global operator given by an integral operator and one looks for a commuting local object given by a second order differential operator Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Setting of the problem in the matrix case “discrete version” of time and band limiting: one deals with a global operator given by a full matrix and one looks for a commuting local object given by a tridiagonal matrix. “continuous version” of time and band limiting: In this talk we deal with a global operator given by an integral operator and one looks for a commuting local object given by a second order differential operator Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
Introduction Matrix Orthogonal Polynomials and band and time limiting Setting of the problem in the matrix case Matrix orthogonal polynomials We consider a sequence of matrix orthonormal polynomials Q n w.r.t a matrix weight W ( t ) , of dimension R , supported in an interval [ a, b ] : � b Q i ( x ) W ( x ) Q j ( x ) ∗ dx = δ ij I � Q i , Q j � W = a Q n are matrix polynomials with non singular leading coefficients Q n ( x ) = A n x n + A n − 1 x n − 1 + . . . + A 0 , A n ∈ M R ( R ) . The sequence Q n is unique up to the multiplication on the left by a unitary matrix. Mirta M. Castro Matrix Orthogonal Polynomials and Time and Band limiting
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