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Plamen Iliev

Publications and source records attributed to Plamen Iliev.

At least 37 records · Page 2Linked to original sources
arXiv2015

Bernstein-Szegő measures, Banach algebras, and scattering theory

We give a simple and explicit description of the Bernstein-Szego type measures associated with Jacobi matrices which differ from the Jacobi matrix of the Chebyshev measure in finitely many entries. We also introduce a class of measures M which parametrizes the Jacobi matrices with exponential decay and for each element in M we define a scattering function. Using Banach algebras associated with increasing Beurling weights, we prove that the exponential decay of the coefficients in a Jacobi matrix is completely determined by the decay of the negative Fourier coefficients of the scattering function. Combining this result with the Bernstein-Szego type measures we provide different characterizations of the rate of decay of the entries of the Jacobi matrices for measures in M.

math.CA↗
arXiv2014

A hypergeometric basis for the Alpert multiresolution analysis

We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$ hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier transform of each basis function is written in terms of ${}_2F_3$ hypergeometric functions. Moreover, the entries in the matrix equation connecting the wavelets with the scaling functions are shown to be balanced ${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute them recursively via three-term recurrence relations. The above results lead to a variety of new interesting identities and orthogonality relations reminiscent to classical identities of higher-order hypergeometric functions and orthogonality relations of Wigner $6j$-symbols.

math.CA↗
arXiv2013

Bispectral extensions of the Askey-Wilson polynomials

Following the pioneering work of Duistermaat and Grünbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one consisting of difference operators acting on the degree index $n$, and another one of operators acting on the variable $x$. The goal of the present paper is to construct and parametrize bispectral extensions of the Askey-Wilson polynomials, where the second algebra consists of $q$-difference operators. In particular, we describe explicitly measures on the real line for which the corresponding orthogonal polynomials satisfy (higher-order) $q$-difference equations extending all known families of orthogonal polynomials satisfying $q$-difference, difference or differential equations in $x$.

math.QA↗
arXiv2013

Polynomials with no zeros on a face of the bidisk

We present a Hilbert space geometric approach to the problem of characterizing the positive bivariate trigonometric polynomials that can be represented as the square of a two variable polynomial possessing a certain stability requirement, namely no zeros on a face of the bidisk. Two different characterizations are given using a Hilbert space structure naturally associated to the trigonometric polynomial; one is in terms of a certain orthogonal decomposition the Hilbert space must possess called the "split-shift orthogonality condition" and another is an operator theoretic or matrix condition closely related to an earlier characterization due to the first two authors. This approach allows several refinements of the characterization and it also allows us to prove a sums of squares decomposition which at once generalizes the Cole-Wermer sums of squares result for two variable stable polynomials as well as a sums of squares result related to the Schur-Cohn method for counting the roots of a univariate polynomial in the unit disk.

math.CV↗
arXiv2012

Fejér-Riesz factorizations and the structure of bivariate polynomials orthogonal on the bi-circle

We give a complete characterization of the positive trigonometric polynomials Q(θ,ϕ) on the bi-circle, which can be factored as Q(θ,ϕ)=|p(e^{iθ},e^{iϕ})|^2 where p(z,w) is a polynomial nonzero for |z|=1 and |w|\leq 1. The conditions are in terms of recurrence coefficients associated with the polynomials in lexicographical and reverse lexicographical ordering orthogonal with respect to the weight 1/(4π^2Q(θ,ϕ)) on the bi-circle. We use this result to describe how specific factorizations of weights on the bi-circle can be translated into identities relating the recurrence coefficients for the corresponding polynomials and vice versa. In particular, we characterize the Borel measures on the bi-circle for which the coefficients multiplying the reverse polynomials associated with the two operators: multiplication by z in lexicographical ordering and multiplication by w in reverse lexicographical ordering vanish after a particular point. This can be considered as a spectral type result analogous to the characterization of the Bernstein-Szegő measures on the unit circle.

math.CV↗
arXiv2012

Meixner polynomials in several variables satisfying bispectral difference equations

We construct a set $M_d$ whose points parametrize families of Meixner polynomials in $d$ variables. There is a natural bispectral involution $b$ on $M_d$ which corresponds to a symmetry between the variables and the degree indices of the polynomials. We define two sets of $d$ commuting partial difference operators diagonalized by the polynomials. One of the sets consists of difference operators acting on the variables of the polynomials and the other one on their degree indices, thus proving their bispectrality. The two sets of partial difference operators are naturally connected via the involution $b$.

math.CA↗
arXiv2011

Orthogonality relations for bivariate Bernstein-Szegő measures

The orthogonality properties of certain subspaces associated with bivariate Bernstein-Szegő measures are considered. It is shown that these spaces satisfy more orthogonality relations than expected from the relations that define them. The results are used to prove a Christoffel-Darboux like formula for these measures.

math.CV↗
arXiv2011

A Lie theoretic interpretation of multivariate hypergeometric polynomials

In 1971 Griffiths used a generating function to define polynomials in d variables orthogonal with respect to the multinomial distribution. The polynomials possess a duality between the discrete variables and the degree indices. In 2004 Mizukawa and Tanaka related these polynomials to character algebras and the Gelfand hypergeometric series. Using this approach they clarified the duality and obtained a new proof of the orthogonality. In the present paper, we interpret these polynomials within the context of the Lie algebra sl_{d+1}. Our approach yields yet another proof of the orthogonality. It also shows that the polynomials satisfy d independent recurrence relations each involving d^2+d+1 terms. This combined with the duality establishes their bispectrality. We illustrate our results with several explicit examples.

math.RT↗
arXiv2011

Krall-Laguerre commutative algebras of ordinary differential operators

In 1999, Grunbaum, Haine and Horozov defined a large family of commutative algebras of ordinary differential operators which have orthogonal polynomials as eigenfunctions. These polynomials are mutually orthogonal with respect to a Laguerre-type weight distribution, thus providing solutions to Krall's problem. In the present paper we give a new proof of their result which establishes a conjecture, concerning the explicit characterization of the dual commutative algebra of eigenvalues. In particular, for the Koornwinder's generalization of Laguerre polynomials, our approach yields an explicit set of generators for the whole algebra of differential operators. We also illustrate how more general Sobolev-type orthogonal polynomials fit within this theory.

math.CA↗
arXiv2011

Krall-Jacobi commutative algebras of partial differential operators

We construct a large family of commutative algebras of partial differential operators invariant under rotations. These algebras are isomorphic extensions of the algebras of ordinary differential operators introduced by Grunbaum and Yakimov corresponding to Darboux transformations at one end of the spectrum of the recurrence operator for the Jacobi polynomials. The construction is based on a new proof of their results which leads to a more detailed description of the one-dimensional theory. In particular, our approach establishes a conjecture by Haine concerning the explicit characterization of the Krall-Jacobi algebras of ordinary differential operators.

math.CA↗
arXiv2010

The Rahman polynomials and the Lie algebra sl_3(C)

We interpret the Rahman polynomials in terms of the Lie algebra $sl_3(C)$. Using the parameters of the polynomials we define two Cartan subalgebras for $sl_3(C)$, denoted $H$ and $\tilde{H}$. We display an antiautomorphism $\dagger$ of $sl_3(C)$ that fixes each element of $H$ and each element of $\tilde{H}$. We consider a certain finite-dimensional irreducible $sl_3(C)$-module $V$ consisting of homogeneous polynomials in three variables. We display a nondegenerate symmetric bilinear form $<,>$ on $V$ such that $<βξ,ζ> = < ξ,β^\dagger ζ>$ for all $β\in sl_3(C)$ and $ξ,ζ\in V$. We display two bases for $V$; one diagonalizes $H$ and the other diagonalizes $\tilde{H}$. Both bases are orthogonal with respect to $<,>$. We show that when $<,>$ is applied to a vector in each basis, the result is a trivial factor times a Rahman polynomial evaluated at an appropriate argument. Thus for both transition matrices between the bases each entry is described by a Rahman polynomial. From these results we recover the previously known orthogonality relation for the Rahman polynomials. We also obtain two seven-term recurrence relations satisfied by the Rahman polynomials, along with the corresponding relations satisfied by the dual polynomials. These recurrence relations show that the Rahman polynomials are bispectral. In our theory the roles of $H$ and $\tilde{H}$ are interchangable, and for us this explains the duality and bispectrality of the Rahman polynomials. We view the action of $H$ and $\tilde{H}$ on $V$ as a rank 2 generalization of a Leonard pair.

math.RT↗
arXiv2009

Bispectral commuting difference operators for multivariable Askey-Wilson polynomials

We construct a commutative algebra A_z, generated by d algebraically independent q-difference operators acting on variables z_1, z_2,..., z_d, which is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that the polynomials P_n(z) possess a certain duality between z and n. Analytic continuation allows us to obtain another commutative algebra A_n, generated by d algebraically independent difference operators acting on the discrete variables n_1, n_2,..., n_d, which is also diagonalized by P_n(z). This leads to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.

math.CA↗
arXiv2009

Bispectrality of multivariable Racah-Wilson polynomials

We construct a commutative algebra A_x of difference operators in R^p, depending on p+3 real parameters which is diagonalized by the multivariable Racah polynomials R_p(n;x) considered by Tratnik [27]. It is shown that for specific values of the variables x=(x_1,x_2,...,x_p) there is a hidden duality between n and x. Analytic continuation allows us to construct another commutative algebra A_n in the variables n=(n_1,n_2,...,n_p) which is also diagonalized by R_p(n;x). Thus R_p(n;x) solve a multivariable discrete bispectral problem in the sense of Duistermaat and Grunbaum [8]. Since a change of the variables and the parameters in the Racah polynomials gives the multivariable Wilson polynomials [26], this change of variables and parameters in A_x and A_n leads to bispectral commutative algebras for the multivariable Wilson polynomials.

math.CA↗
arXiv2008

Trigonometric Darboux transformations and Calogero-Moser matrices

We characterize in terms of Darboux transformations the spaces in the Segal-Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of e^x. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero-Moser matrices.

math.QA↗
arXiv2008

On a two variable class of Bernstein-Szego measures

The one variable Bernstein-Szego theory for orthogonal polynomials on the real line is extended to a class of two variable measures. The polynomials orthonormal in the total degree ordering and the lexicographical ordering are constructed and their recurrence coefficients discussed.

math.CA↗
arXiv2007

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation $u_t=Lu$ where $L$ is a second-order difference operator in a discrete variable $n$. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients $α_k(n,m)$ in this expansion are analogs of Hadamard's coefficients for the (continuous) Schrodinger operator. We derive an explicit formula for $α_k$ in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals $n=m$ and $n=m+1$ define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable $t$, if and only if the operator $L$ belongs to the family of bispectral operators constructed in [18].

math-ph↗
arXiv2007

Rational Ruijsenaars-Schneider hierarchy and bispectral difference operators

We show that a monic polynomial in a discrete variable $n$, with coefficients depending on time variables $t_1, t_2,...$ is a $τ$-function for the discrete Kadomtsev-Petviashvili hierarchy if and only if the motion of its zeros is governed by a hierarchy of Ruijsenaars-Schneider systems. These $τ$-functions were considered in [12], where it was proved that they parametrize rank one solutions to a difference-differential version of the bispectral problem.

math-ph↗