Searcharxiv⌕ Search

arXiv subjects

F. Alberto Grünbaum

Publications and source records attributed to F. Alberto Grünbaum.

At least 19 recordsLinked to original sources
arXiv2025

The relative importance of being Gaussian

The remarkable results for denoising in computer vision using diffusion models given in \cite{SDWMG,HJA,HHG} yield a robust mathematical justification for algorithms based on crucial properties of a sequence of Gaussian independent $N(0,1)$ random variables. In particular the derivations use the fact that a Gaussian distribution is determined by its mean and variance and that the sum of two Gaussians is another Gaussian. \bigskip The issue raised in this short note is the following: suppose we use the algorithm without any changes but replace the nature of the noise and use, for instance, uniformly distributed noise or noise with a Beta distribution, or noise which is a random superposition of two Gaussians with very different variances. One could, of course, try to modify the algorithm keeping in mind the nature of the noise, but this is not what we do. Instead we study the performance of the algorithm when used with noise that is very far in nature from the Gaussian case, where it is designed to work well. Usually these algorithms are implemented on very powerful computers. Our experiments are all carried out on a small laptop and for the smallest possible image size. Exploring how our observations are confirmed or changed when dealing in different situations remains an interesting challenge.

cs.CV↗
arXiv2022

Matrix Bispectrality and Noncommutative Algebras: beyond the prolate spheroidals

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 operator with a simple spectrum in its commutator. In this article, we discuss a noncommutative version of the bispectral problem, obtained by allowing all objects in the original formulation to be matrix-valued. Deep attention is given to bispectral algebras and their presentations as a tool to get information about bispectral triples.

math.FA↗
arXiv2021

An urn model for the Jacobi-Piñeiro polynomials

The list of physically motivated urn models that can be solved in terms of classical orthogonal polynomials is very small. It includes a model proposed by D. Bernoulli and further analyzed by S. Laplace and a model proposed by P. and T. Ehrenfest and eventually connected with the Krawtchouk and Hahn polynomials. This connection was reversed recently in the case of the Jacobi polynomials where a rather contrived, and later a simpler urn model was proposed. Here we consider an urn model going with the Jacobi-Piñeiro multiple orthogonal polynomials. These polynomials have recently been put forth in connection with a stochastic matrix.

math.PR↗
arXiv2020

Reflective prolate-spheroidal operators and the adelic Grassmannian

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. Previously, such pairs were constructed by ad hoc methods, which worked because a commuting operator of low order could be found by a direct calculation. We describe a general approach to these problems that proves that every point $W$ of Wilson's infinite dimensional adelic Grassmannian $\mathrm Gr^ad$ gives rise to an integral operator $T_W$, acting on $L^2(Γ)$ for a contour $Γ\subset\mathbb C$, which reflects a differential operator $R(z,\partial_z)$ in the sense that $R(-z,-\partial_z)\circ T_W=T_W\circ R(w,\partial_w)$ on a dense subset of $L^2(Γ)$. By using analytic methods and methods from integrable systems, we show that the reflected differential operator can be constructed from the Fourier algebra of the associated bispectral function $ψ_W(x,z)$. The size of this algebra with respect to a bifiltration is in turn determined using algebro-geometric methods. Intrinsic properties of four involutions of the adelic Grassmannian naturally lead us to consider the reflecting property in place of plain commutativity. Furthermore, we prove that the time-band limited operators of the generalized Laplace transforms with kernels given by all rank one bispectral functions $ψ_W(x,-z)$ reflect a differential operator. A $90^\circ$ rotation argument is used to prove that the time-band limited operators of the generalized Fourier transforms with kernels $ψ_W(x,iz)$ admit a commuting differential operator. These methods produce vast collections of integral operators with prolate-spheroidal properties, associated to the wave functions of all rational solutions of the KP hierarchy vanishing at infinity, introduced by Krichever in the late 1970s.

math.CA↗
arXiv2018

A Feyman-Kac approach to a paper of Chung and Feller on fluctuations in the coin-tossing game

A classical result of K. L. Chung and W. Feller deals with the partial sums $S_k$ arising in a fair coin-tossing game. If $N_n$ is the number of "positive" terms among $S_1, S_2,\dots,S_n$ then the quantity $P(N_{2n}=2r)$ takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for $P(N_{2n+1}=r)$, $r=0,1,2,\dots,2n+1$. We get to this result by adapting the Feynman-Kac methodology.

math.PR↗
arXiv2018

Bispectrality and Time-Band-Limiting: Matrix valued polynomials

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.

math.CA↗
arXiv2017

Algebraic Heun operator and band-time limiting

We introduce the algebraic Heun operator associated to any bispectral pair of operators. We show that these operators are natural generalizations of the ordinary Heun operator. This leads to a simple construction of the operators commuting with the projection operators in problems of band-time limiting and it gives a way to adapt a construction first used by Perline in a purely finite setup to quite a few other situations. We also extend his algebraic construction to cover some purely finite cases where his proposal fails to give a useful commuting operator.

math-ph↗
arXiv2017

Stochastic LU factorizations, Darboux transformations and urn models

We consider UL (and LU) decompositions of the one-step transition probability matrix of a random walk with state space the nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We give conditions on the free parameter of the UL factorization in terms of certain continued fraction such that this stochastic factorization is possible. By inverting the order of multiplication (also known as a Darboux transformation) we get a new family of random walks where it is possible the identify the spectral measures in terms of a Geronimus transformation. The same can be done for the LU factorization but now without a free parameter. Finally, we apply our results in two examples, the random walk with constant transition probabilities and the random walk generated by the Jacobi orthogonal polynomials. In both situations we give urn models associated with all the random walks in question.

math.PR↗
arXiv2017

A generalization of Schur functions: applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks

Recent work on recurrence in quantum walks has provided a representation of Schur functions in terms of unitary operators. We propose a generalization of Schur functions by extending this operator representation to arbitrary operators on Banach spaces. Such generalized Schur functions meet the formal structure of first return generating functions, thus we call them FR-functions. We derive general properties of FR-functions, among them a generalization of the renewal equation already known for random and quantum walks, as well as splitting properties which extend useful factorizations of Schur functions. When specialized to self-adjoint operators, FR-functions become Nevanlinna functions. This leads to new results on Nevanlinna functions: decomposition rules which are the analogue of useful factorizations of Schur functions, a simple Nevanlinna version of the Schur algorithm and new operator and integral representations of Nevanlinna functions which, in contrast to standard ones, are exact analogues of those already known for Schur functions, giving similarly a one-to-one correspondence between Nevanlinna functions and measures on the real line. The paper is completed with several applications of FR-functions to orthogonal polynomials and random and quantum walks which illustrate their wide interest: an analogue for orthogonal polynomials on the real line of the Khrushchev formula for orthogonal polynomials on the unit circle, and the use of FR-functions to study recurrence in random walks, quantum walks and open quantum walks. These applications provide numerous explicit examples of FR-functions and their splittings and show that these new tools, despite being extensions of very classical ones, play an important role in the study of physical problems of a highly topical nature.

math.FA↗
arXiv2016

Tridiagonalization and the Heun equation

It is shown that the tridiagonalization of the hypergeometric operator $L$ yields the generic Heun operator $M$. The algebra generated by the operators $L,M$ and $Z=[L,M]$ is quadratic and a one-parameter generalization of the Racah algebra. The new Racah-Heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators $L$ and $M$. An interpretation in terms of the Racah problem for $su(1,1)$ algebras and separation of variables in a superintegrable system are discussed.

math-ph↗
arXiv2015

Time and Band Limiting for Matrix Valued Functions, an Example

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 foundations of information theory and to a remarkable series of papers by D. Slepian, H. Landau and H. Pollak. To our knowledge, this is the first example showing in a non-commutative setup that a bispectral property implies that the corresponding global operator of "time and band limiting" admits a commuting local operator. This is a noncommutative analog of the famous prolate spheroidal wave operator.

math.SP↗
arXiv2015

A limit theorem for a 3-period time-dependent quantum walk

We consider a discrete-time 2-state quantum walk on the line. The state of the quantum walker evolves according to a rule which is determined by a coin-flip operator and a position-shift operator. In this paper we take a 3-periodic time evolution as the rule. For such a quantum walk, we get a limit distribution which expresses the asymptotic behavior of the walker after a long time. The limit distribution is different from that of a time-independent quantum walk or a 2-period time-dependent quantum walk. We give some analytical results and then consider a number of variants of our model and indicate the result of simulations for these ones.

quant-ph↗
arXiv2014

Some Noncommutative Matrix Algebras Arising in the Bispectral Problem

I revisit the so called "bispectral problem" introduced in a joint paper with Hans Duistermaat a long time ago, allowing now for the differential operators to have matrix coefficients and for the eigenfunctions, and one of the eigenvalues, to be matrix valued too. In the last example we go beyond this and allow both eigenvalues to be matrix valued.

math.SP↗
arXiv2013

On a Seminal Paper by Karlin and McGregor

The use of spectral methods to study birth-and-death processes was pioneered by S. Karlin and J. McGregor. Their expression for the transition probabilities was made explicit by them in a few cases. Here we complete their analysis and indicate a few applications of their very powerful method.

math.PR↗
arXiv2011

A System of Multivariable Krawtchouk Polynomials and a Probabilistic Application

The one variable Krawtchouk polynomials, a special case of the $_2F_1$ function did appear in the spectral representation of the transition kernel for a Markov chain studied a long time ago by M. Hoare and M. Rahman. A multivariable extension of this Markov chain was considered in a later paper by these authors where a certain two variable extension of the $F_1$ Appel function shows up in the spectral analysis of the corresponding transition kernel. Independently of any probabilistic consideration a certain multivariable version of the Gelfand-Aomoto hypergeometric function was considered in papers by H. Mizukawa and H. Tanaka. These authors and others such as P. Iliev and P. Tertwilliger treat the two-dimensional version of the Hoare-Rahman work from a Lie-theoretic point of view. P. Iliev then treats the general $n$-dimensional case. All of these authors proved several properties of these functions. Here we show that these functions play a crucial role in the spectral analysis of the transition kernel that comes from pushing the work of Hoare-Rahman to the multivariable case. The methods employed here to prove this as well as several properties of these functions are completely different to those used by the authors mentioned above.

math.PR↗
arXiv2011

Properties of Matrix Orthogonal Polynomials via their Riemann-Hilbert Characterization

We give a Riemann-Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, something that is not possible in the scalar case. The combination of the ladder operators will lead to a family of second-order differential equations satisfied by the orthogonal polynomials, some of them of 0-th and first order, something also impossible in the scalar setting. This shows that the differential properties in the matrix case are much more complicated than in the scalar situation. We will study several examples given in the last years as well as others not considered so far.

math.CA↗