On Generalized Gaussian Quadratures for Exponentials and Their Applications

Abstract We introduce new families of Gaussian-type quadratures for weighted integrals of exponential functions and consider their applications to integration and interpolation of bandlimited functions. We use a generalization of a representation theorem due to Caratheodory to derive these quadratures. For each positive measure, the quadratures are parameterized by eigenvalues of the Toeplitz matrix constructed from the trigonometric moments of the measure. For a given accuracy ϵ, selecting an eigenvalue close to ϵ yields an approximate quadrature with that accuracy. To compute its weights and nodes, we present a new fast algorithm. These new quadratures can be used to approximate and integrate bandlimited functions, such as prolate spheroidal wave functions, and essentially bandlimited functions, such as Bessel functions. We also develop, for a given precision, an interpolating basis for bandlimited functions on an interval.

[1]  D. Slepian.  Prolate spheroidal wave functions, fourier analysis, and uncertainty — V: the discrete case , 1978, The Bell System Technical Journal.

[2]  Georg Heinig,et al.  Algebraic Methods for Toeplitz-like Matrices and Operators , 1984 .

[3]  Norman Yarvin,et al.  Generalized Gaussian Quadratures and Singular Value Decompositions of Integral Operators , 1998, SIAM J. Sci. Comput..

[4]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[5]  Hao Lu.  Fast Solution of Confluent Vandermonde Linear Systems , 1994 .

[6]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

[7]  D. Slepian.  Some Asymptotic Expansions for Prolate Spheroidal Wave Functions , 1965 .

[8]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[9]  H. Landau.  The eigenvalue behavior of certain convolution equations , 1965 .

[10]  W. J. Studden,et al.  Tchebycheff Systems: With Applications in Analysis and Statistics. , 1967 .

[11]  G. Beylkin.  On the Fast Fourier Transform of Functions with Singularities , 1995 .

[12]  W. J. Thron,et al.  Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle , 1989 .

[13]  A.,et al.  FAST FOURIER TRANSFORMS FOR NONEQUISPACED DATA * , 2022 .

[14]  L. Greengard,et al.  Regular Article: A Fast Adaptive Multipole Algorithm in Three Dimensions , 1999 .

[15]  Inversion formulas and fast algorithms for Löwner-Vandermonde matrices , 1998 .

[16]  Israel Gohberg,et al.  Circulants, displacements and decompositions of matrices , 1992 .

[17]  H. Montgomery.  Ten lectures on the interface between analytic number theory and harmonic analysis , 1994 .

[18]  V. Pisarenko.  The Retrieval of Harmonics from a Covariance Function , 1973 .

[19]  William F. Trench,et al.  Some Spectral Properties of Hermitian Toeplitz Matrices , 1994 .

[20]  Paul Turán,et al.  On a new method of analysis and its applications , 1984 .

[21]  V. Rokhlin,et al.  Prolate spheroidal wavefunctions, quadrature and interpolation , 2001 .

[22]  U. Grenander,et al.  Toeplitz Forms And Their Applications , 1958 .

[23]  Y. Genin,et al.  Spectral properties of finite Toeplitz matrices , 1984 .

[24]  D. Slepian,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — II , 1961 .

[25]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[26]  Norman Yarvin,et al.  An Improved Fast Multipole Algorithm for Potential Fields on the Line , 1999 .