Time and band limiting for matrix valued functions: an integral and a commuting differential operator

AbstractIn the mid 1980s it was conjectured that every bispectral meromorphic function {\psi(x,y)} gives rise to an integral operator {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions {\psi(x,y)} where the commuting differential operator is of order {\leq 6}. We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

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Bispectrality and Time–Band Limiting: Matrix-valued Polynomials

International Mathematics Research Notices ◽

10.1093/imrn/rny140 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 4016-4036 ◽

Cited By ~ 1

Author(s):

F Alberto Grünbaum ◽

Inés Pacharoni ◽

Ignacio Zurrián

Keyword(s):

Signal Processing ◽

Differential Operator ◽

Integral Operator ◽

General Condition ◽

Integrable Systems ◽

Joint Work ◽

The Subject ◽

Time Band ◽

Commuting Differential Operator

Abstract The subject of time–band limiting, originating in signal processing, is dominated by the miracle that a naturally appearing integral operator admits a commuting differential one allowing for a numerically efficient way to compute its eigenfunctions. Bispectrality is an effort to dig into the reasons behind this miracle and goes back to joint work with H. Duistermaat. This search has revealed unexpected connections with several parts of mathematics, including integrable systems. Here we consider a matrix-valued version of bispectrality and give a general condition under which we can display a constructive and simple way to obtain the commuting differential operator. Furthermore, we build an operator that commutes with both the time-limiting operator and the band-limiting operators.

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Revisiting the Concentration Problem of Vector Fields within a Spherical Cap: A Commuting Differential Operator Solution

Journal of Fourier Analysis and Applications ◽

10.1007/s00041-014-9324-7 ◽

2014 ◽

Vol 20 (2) ◽

pp. 421-451 ◽

Cited By ~ 9

Author(s):

Kornél Jahn ◽

Nándor Bokor

Keyword(s):

Differential Operator ◽

Vector Fields ◽

Spherical Cap ◽

Operator Solution ◽

Concentration Problem ◽

Commuting Differential Operator

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On guaranteed eigenvalue estimation of compact differential operator with singularity

IEICE Proceeding Series ◽

10.15248/proc.1.812 ◽

2014 ◽

Vol 1 ◽

pp. 812-815

Author(s):

Xuefeng LIU ◽

Shin'ich OISHI

Keyword(s):

Differential Operator ◽

Eigenvalue Estimation

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Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2006.11.1.14764 ◽

2006 ◽

Vol 11 (1) ◽

pp. 47-78 ◽

Cited By ~ 4

Author(s):

S. Pečiulytė ◽

A. Štikonas

Keyword(s):

Boundary Condition ◽

Differential Operator ◽

Boundary Conditions ◽

Nonlocal Boundary Condition ◽

Nonlocal Boundary ◽

Liouville Problem ◽

Complex Case ◽

Qualitative Behavior ◽

Point Boundary ◽

Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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