arXiv subjects

Ryszard Szwarc

Publications and source records attributed to Ryszard Szwarc.

At least 19 recordsLinked to original sources

arXiv2026

Indeterminate Jacobi operators II

We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, and present countable subsets S of the domain D(T) such that span(S) is dense in \ell^2. As an example we have S={(p_n(u))+B(u)(p_n(0)):D(u)=0, u\neq 0}, where (p_n) denotes the orthonormal polynomials of the moment problem and B,D are two of the Nevanlinna functions. It is also proved that sets like S are optimal in the sense that if one vector is removed, then the span is no longer dense.

math.FA↗

arXiv2026

Special N-extremal solutions to indeterminate moment problems

For an N-extremal solution $μ$ to an indeterminate moment problem it is known by a theorem of M. Riesz that the measure $(1+x^2)^{-1}dμ(x)$ is determinate. For $0<α<1$ we show by contradiction that there exist indeterminate N-extremal solutions $μ$ such that $(1+x^2)^{-α}dμ(x)$ is determinate, and there exist also indeterminate N-extremal solutions $μ$ such that $(1+x^2)^{-α}dμ(x)$ is indeterminate. Explicit examples of such measures are so far only known when $α=1/2$. For indeterminate Stieltjes moment problems and for N-extremal solutions $μ$, we show that $(1+x^2)^{-1/2}dμ(x)$ is indeterminate except when $μ=μ_F$ is the Friedrichs solution in case of which $(1+x^2)^{-1/2}dμ_F(x)$ is determinate. We identify the Friedrichs and Krein solutions for some indeterminate Stieltjes moment problems.

math.FA↗

arXiv2023

Dual spaces vs. Haar measures of polynomial hypergroups

Many symmetric orthogonal polynomials $(P_n(x))_{n\in\mathbb{N}_0}$ induce a hypergroup structure on $\mathbb{N}_0$. The Haar measure is the counting measure weighted with $h(n):=1/\int_\mathbb{R}\!P_n^2(x)\,\mathrm{d}μ(x)\geq1$, where $μ$ denotes the orthogonalization measure. We observed that many naturally occurring examples satisfy the remarkable property $h(n)\geq2\;(n\in\mathbb{N})$. We give sufficient criteria and particularly show that $h(n)\geq2\;(n\in\mathbb{N})$ if the (Hermitian) dual space $\widehat{\mathbb{N}_0}$ equals the full interval $[-1,1]$, which is fulfilled by an abundance of examples. We also study the role of nonnegative linearization of products (and of the harmonic and functional analysis resulting from such expansions). Moreover, we construct two example types with $h(1)<2$. To our knowledge, these are the first such examples. The first type is based on Karlin-McGregor polynomials, and $\widehat{\mathbb{N}_0}$ consists of two intervals and can be chosen "maximal" in some sense; $h$ is of quadratic growth. The second type relies on certain compact operators; $h$ grows exponentially, and $\widehat{\mathbb{N}_0}$ is discrete.

math.CA↗

arXiv2023

Indeterminate Jacobi operators

We consider the Jacobi operator (T,D(T)) associated with an indeterminate Hamburger moment problem, i.e., the operator in $\ell^2$ defined as the closure of the Jacobi matrix acting on the subspace of complex sequences with only finitely many non-zero terms. It is well-known that it is symmetric with deficiency indices (1,1). For a complex number z let $\mathfrak{p}_z, \mathfrak{q}_z$ denote the square summable sequences (p_n(z)) and (q_n(z)) corresponding to the orthonormal polynomials p_n and polynomials q_n of the second kind. We determine whether linear combinations of $\mathfrak{p}_u,\mathfrak{p}_v,\mathfrak{q}_u,\mathfrak{q}_v$ for complex u,v belong to D(T) or to the domain of the self-adjoint extensions of T in $\ell^2$. The results depend on the four Nevanlinna functions of two variables associated with the moment problem. We also show that D(T) is the common range of an explicitly constructed family of bounded operators on $\ell^2$.

math.FA↗

arXiv2022

Self-adjoint operators associated with Hankel moment matrices

In a paper from 2016 D. R. Yafaev initiated a study of closable Hankel forms associated with the moments $(m_n)$ of a positive measure with infinite support on the real line. If $m_n=o(1)$ Yafaev characterized the closure of the form based on earlier work on quasi-Carleman operators. We give a new proof of the description of the closure based entirely on moment considerations. The main purpose of the present paper is a description of the self-adjoint Hankel operators associated with closed Hankel forms in the Hilbert space of square summable sequences. We do this not only in the case $m_n=o(1)$ studied by Yafaev but also in two other cases, where the Hankel form is closable, namely if the moment sequence is indeterminate or if the moment sequence is determinate with finite index of determinacy.

math.FA↗

arXiv2021

Positivity of Turán determinants for orthogonal polynomials II

The polynomials $p_n$ orthogonal on the interval $[-1,1],$ normalized by $p_n(1)=1,$ satisfy Turán's inequality if $p_n^2(x)-p_{n-1}(x)p_{n+1}(x)\ge 0$ for $n\ge 1$ and for all $x$ in the interval of orthogonality. We give a general criterion for orthogonal polynomials to satisfy Turán's inequality. This extends essentially the results of \cite{szw}. In particular the results can be applied to many classes of orthogonal polynomials, by inspecting their recurrence relation.

math.CA↗

arXiv2019

Closable Hankel operators and moment problems

In a paper from 2016 D. R. Yafaev considers Hankel operators associated with Hamburger moment sequences q_n and claims that the corresponding Hankel form is closable if and only if the moment sequence tends to 0. The claim is not correct, since we prove closability for any indeterminate moment sequence but also for certain determinate moment sequences corresponding to measures with finite index of determinacy. It is also established that Yafaev's result holds if the moments satisfy \root{2n}\of{q_{2n}}=o(n).

math.FA↗

arXiv2019

On a property of random walk polynomials involving Christoffel functions

Discrete-time birth-death processes may or may not have certain properties known as asymptotic aperiodicity and the strong ratio limit property. In all cases known to us a suitably normalized process having one property also possesses the other, suggesting equivalence of the two properties for a normalized process. We show that equivalence may be translated into a property involving Christoffel functions for a type of orthogonal polynomials known as random walk polynomials. The prevalence of this property - and thus the equivalence of asymptotic aperiodicity and the strong ratio limit property for a normalized birth-death process - is proven under mild regularity conditions.

math.CA↗

arXiv2018

Inverse of Infinite Hankel Moment Matrices

Let $(s_n)_{n\ge 0}$ denote an indeterminate Hamburger moment sequence and let $\mathcal H=\{s_{m+n}\}$ be the corresponding positive definite Hankel matrix. We consider the question if there exists an infinite symmetric matrix $\mathcal A=\{a_{j,k}\}$, which is an inverse of $\mathcal H$ in the sense that the matrix product $\mathcal A\mathcal H$ is defined by absolutely convergent series and $\mathcal A\mathcal H$ equals the identity matrix $\mathcal I$, a property called (aci). A candidate for $\mathcal A$ is the coefficient matrix of the reproducing kernel of the moment problem, considered as an entire function of two complex variables. We say that the moment problem has property (aci), if (aci) holds for this matrix $\mathcal A$. We show that this is true for many classical indeterminate moment problems but not for the symmetrized version of a cubic birth-and-death process studied by Valent and co-authors. We consider mainly symmetric indeterminate moment problems and give a number of sufficient conditions for (aci) to hold in terms of the recurrence coefficients for the orthonormal polynomials. A sufficient condition is a rapid increase of the recurrence coefficients in the sense that the quotient between consecutive terms is uniformly bounded by a constant strictly smaller than one. We also give a simple example, where (aci) holds, but an inverse matrix of $\mathcal H$ is highly non-unique.

math.CA↗

arXiv2017

Symmetric moment problems and a conjecture of Valent

In 1998 G. Valent made conjectures about the order and type of certain indeterminate Stieltjes moment problems associated with birth and death processes having polynomial birth and death rates of degree p\ge 3. Romanov recently proved that the order is 1/p as conjectured, see \cite{Ro}. We prove that the type with respect to the order is related to certain multi-zeta values and that this type belongs to the interval [π/(p\sin(π/p)),π/(p\sin(π/p)\cos(π/p))], which also contains the conjectured value. This proves that the conjecture about type is asymptotically correct as p\to\infty. The main idea is to obtain estimates for order and type of symmetric indeterminate Hamburger moment problems when the orthonormal polynomials P_n and those of the second kind Q_n satisfy P_{2n}^2(0)\sim c_1n^{-1/\b} and Q_{2n-1}^2(0)\sim c_2 n^{-1/\a}, where 0<\a,\b<1 can be different, and c_1,c_2 are positive constants. In this case the order of the moment problem is majorized by the harmonic mean of \a,\b. Here α_n\sim β_n means that α_n/β_n\to 1. This also leads to a new proof of Romanov's Theorem that the order is 1/p.

math.CA↗

arXiv2016

Inhomogeneous Jacobi matrices on trees

We study Jacobi matrices on trees with one end at inifinity. We show that the defect indices cannot be greater than 1 and give criteria for essential selfadjointness. We construct certain polynomials associated with matrices, which mimic orthogonal polynomials in the classical case. Nonnegativity of Jacobi matrices is studied as well.

math.FA↗

arXiv2014

A determinant characterization of moment sequences with finitely many mass-points

To a sequence (s_n)_{n\ge 0} of real numbers we associate the sequence of Hankel matrices \mathcal H_n=(s_{i+j}),0\le i,j \le n. We prove that if the corresponding sequence of Hankel determinants D_n=\det\mathcal H_n satisfy D_n>0 for n<n_0 while D_n=0 for n\ge n_0, then all Hankel matrices are positive semi-definite, and in particular (s_n) is the sequence of moments of a discrete measure concentrated in n_0 points on the real line. We stress that the conditions D_n\ge 0 for all n do not imply the positive semi-definiteness of the Hankel matrices.

math.CA↗

arXiv2013

On the order of indeterminate moment problems

For an indeterminate moment problem we denote the orthonormal polynomials by P_n. We study the relation between the growth of the function P(z)=(\sum_{n=0}^\infty|P_n(z)|^2)^{1/2} and summability properties of the sequence (P_n(z)). Under certain assumptions on the recurrence coefficients from the three term recurrence relation zP_n(z)=b_nP_{n+1}(z)+a_nP_n(z)+b_{n-1}P_{n-1}(z), we show that the function P is of order αwith 0<α<1, if and only if the sequence (P_n(z)) is absolutely summable to any power greater than 2α. Furthermore, the order αis equal to the exponent of convergence of the sequence (b_n). Similar results are obtained for logarithmic order and for more general types of slow growth. To prove these results we introduce a concept of an order function and its dual. We also relate the order of P with the order of certain entire functions defined in terms of the moments or the leading coefficient of P_n

math.CA↗

arXiv2009

Schur Multipliers and Spherical Functions on Homogeneous Trees

Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on the p-adic group PGL_2(Q_q) for every prime number q.

math.GR↗

arXiv2009

Jacobi matrices on trees

Symmetric Jacobi matrices on one sided homogeneous trees are studied. Essential selfadjointness of these matrices turns out to depend on the structure of the tree. If a tree has one end and infinitely many origin points the matrix is always essentially selfadjoint independently of the growth of its coefficients. In case a tree has one origin and infinitely many ends, the essential selfadjointness is equivalent to that of an ordinary Jacobi matrix obtained by the restriction to the so called radial functions. For nonselfadjoint matrices the defect spaces are described in terms of the Poisson kernel associated with the boundary of the tree.

math.FA↗

arXiv2009

The smallest eigenvalue of Hankel matrices

Let H_N=(s_{n+m}),n,m\le N denote the Hankel matrix of moments of a positive measure with moments of any order. We study the large N behaviour of the smallest eigenvalue lambda_N of H_N. It is proved that lambda_N has exponential decay to zero for any measure with compact support. For general determinate moment problems the decay to 0 of lambda_N can be arbitrarily slow or arbitrarily fast. In the indeterminate case, where lambda_N is known to be bounded below by a positive constant, we prove that the limit of the n'th smallest eigenvalue of H_N for N tending to infinity tends rapidly to infinity with n. The special case of the Stieltjes-Wigert polynomials is discussed.

math.CA↗

arXiv2007

Bounds on Tur{á}n determinants

Let μdenote a symmetric probability measure on [-1,1] and let (p_n) be the corresponding orthogonal polynomials normalized such that p_n(1)=1. We prove that the normalized Tur{á}n determinant Δ_n(x)/(1-x^2), where Δ_n=p_n^2-p_{n-1}p_{n+1}, is a Tur{á}n determinant of order n-1 for orthogonal polynomials with respect to (1-x^2)dμ(x). We use this to prove lower and upper bounds for the normalized Tur{á}n determinant in the interval -1<x<1.

math.CA↗

arXiv2007

The ratio and generating function of cogrowth coefficients of finitely generated groups

Let G be a group generated by $r$ elements $g_1,g_2,..., g_r.$ Among the reduced words in $g_1,g_2,..., g_r$ of length $n$ some, say $γ_n,$ represent the identity element of the group $G.$ It has been shown in a combinatorial way that the $2n$th root of $γ_{2n}$ has a limit, called the cogrowth exponent with respect to generators $g_1,g_2,..., g_r.$ We show by analytic methods that the numbers $γ_n$ vary regularly; i.e. the ratio $γ_{2n+2}/γ_{2n}$ is also convergent. Moreover we derive new precise information on the domain of holomorphy of $γ(z),$ the generating function associated with the coefficients $γ_n.$

math.FA↗