Related papers: Matrix valued discrete-continuous functions with t…
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix…
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…
We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference…
In this paper we describe a remarkable new property of the self-adjoint extension W of the prolate spheroidal operator introduced in \cite{college98},\cite{CMbook}. The restriction of this operator to the interval J whose characteristic…
Landau, Pollak, Slepian, and Tracy, Widom discovered that certain integral operators with so called Bessel and Airy kernels possess commuting differential operators and found important applications of this phenomena in time-band limiting…
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined…
We construct families of bispectral difference operators of the form a(n)T + b(n) + c(n) T^{-1} where T is the shift operator. They are obtained as discrete Darboux transformations from appropriate extensions of Jacobi operators. We…
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…
Commuting integral and differential operators connect the topics of Signal Processing, Random Matrix Theory, and Integrable Systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special…
The prolate spheroidal wave functions, which are a special case of the spheroidal wave functions, possess a very surprising and unique property [6]. They are an orthogonal basis of both $L^2(-1,1)$ and the Paley-Wiener space of bandlimited…
New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point…
Bandlimiting and timelimiting operators play a fundamental role in analyzing bandlimited signals that are approximately timelimited (or vice versa). In this paper, we consider a time-frequency (in the discrete Fourier transform (DFT)…
Prolate spheroidal wave functions (PSWFs) play an important role in various areas, from physics (e.g. wave phenomena, fluid dynamics) to engineering (e.g. signal processing, filter design). Even though the significance of PSWFs was realized…
The bispectral problem is motivated by an effort to understand and extend a remarkable phenomenon in Fourier analysis on the real line: the operator of time-and-band limiting is an integral operator admitting a second-order differential…
In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the…
In the mid 80's 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…
We show that spheroidal wave functions viewed as the essential part of the joint eigenfunction of two commuting operators of $L_2(S^2)$ has a defect in the joint spectrum that makes a global labelling of the joint eigenfunctions by quantum…
Beginning with the work of Landau, Pollak and Slepian in the 1960s on time-band limiting, commuting pairs of integral and differential operators have played a key role in signal processing, random matrix theory and integrable systems.…
Replacing operators with continuous operator-valued functions, we prove time-dependent versions of well-known results on compressions and diagonals of bounded operators. The setting of smooth functions is also addressed. Our results have no…
We explore the connections between singular Weyl-Titchmarsh theory and the single and double commutation methods. In particular, we compute the singular Weyl function of the commuted operators in terms of the original operator. We apply the…