On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

On classical orthogonal polynomials

Following the works of Nikiforov and Uvarov a review of the hypergeometric-type difference equation for a functiony(x(s)) on a nonuniform latticex(s) is given. It is shown that the difference-derivatives ofy(x(s)) also satisfy similar equations, if and only ifx(s) is a linear,q-linear, quadratic, or aq-quadratic lattice. This characterization is then used to give a definition of classical orthogonal polynomials, in the broad sense of Hahn, and consistent with the latest definition proposed by Andrews and Askey. The rest of the paper is concerned with the details of the solutions: orthogonality, boundary conditions, moments, integral representations, etc. A classification of classical orthogonal polynomials, discrete as well as continuous, on the basis of lattice type, is also presented.     

On Kernel polynomials and self-perturbation of orthogonal polynomials

Given an orthogonal polynomial system {Q _n (x)} _n=0 ^∞, define another polynomial system by where α _n are complex numbers and t is a positive integer. We find conditions for {P _n (x)} _n=0 ^∞ to be an orthogonal polynomial system. When t=1 and α₁≠0, it turns out that {Q _n (x)} _n=0 ^∞ must be kernel polynomials for {P _n (x)} _n=0 ^∞ for which we study, in detail, the location of zeros and semi-classical character. Received: November 25, 1999; in final form: April 6, 2000?Published online: June 22, 2001     

On the orthogonal polynomials in several variables

We study the connection between orthogonal polynomials in several variables and families of commuting symmetric operators of a special form.     

On sequences of Hurwitz polynomials related to orthogonal polynomials

ABSTRACT

In this contribution, we explore the well-known connection between Hurwitz and orthogonal polynomials. Namely, given a Hurwitz polynomial, it is shown that it can be decomposed into two parts: a polynomial that is orthogonal with respect to some positive measure supported in the positive real axis and its corresponding second-kind polynomial. Conversely, given a sequence of orthogonal polynomials with respect to a positive measure supported in the positive real axis, a sequence of Hurwitz polynomials can be constructed. Based on that connection, we construct sequences of Hurwitz polynomials that satisfy a recurrence relation, in a similar way as the orthogonal polynomials do. Even more, we present a way to construct families of Hurwitz polynomials using two sequences of parameters and a recurrence relation that constitutes an analogue of Favard's theorem in the theory of orthogonal polynomials.     

On zeros of discrete orthogonal polynomials

We exploit difference equations to establish sharp inequalities on the extreme zeros of the classical discrete orthogonal polynomials, Charlier, Krawtchouk, Meixner and Hahn. We also provide lower bounds on the minimal distance between their consecutive zeros.     

Combinatorial orthogonal expansions

The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.

     

On linearly related orthogonal polynomials in several variables

Let \(\{\mathbb{P}_{n}\}_{n\ge 0}\) and \(\{\mathbb{Q}_{n}\}_{n\ge 0}\) be two monic polynomial systems in several variables satisfying the linear structure relation \(\mathbb{Q}_{n} = \mathbb{P}_{n} + M_{n} \mathbb{P}_{n-1}, \quad n\ge 1,\) where M _n are constant matrices of proper size and \(\mathbb{Q}_{0} = \mathbb{P}_{0}\) . The aim of our work is twofold. First, if both polynomial systems are orthogonal, characterize when that linear structure relation exists in terms of their moment functionals. Second, if one of the two polynomial systems is orthogonal, study when the other one is also orthogonal. Finally, some illustrative examples are presented.     

On Koornwinder classical orthogonal polynomials in two variables

In 1975, Tom Koornwinder studied examples of two variable analogues of the Jacobi polynomials in two variables. Those orthogonal polynomials are eigenfunctions of two commuting and algebraically independent partial differential operators. Some of these examples are well known classical orthogonal polynomials in two variables, such as orthogonal polynomials on the unit ball, on the simplex or the tensor product of Jacobi polynomials in one variable, but the remaining cases are not considered classical by other authors. The definition of classical orthogonal polynomials considered in this work provides a different perspective on the subject. We analyze in detail Koornwinder polynomials and using the Koornwinder tools, new examples of orthogonal polynomials in two variables are given.     

On biorthogonal systems associated to orthogonal polynomials

We consider biorthogonal systems of functions associated to derivatives of orthogonal polynomials in the case of general weights. For Freud polynomials, it is proved that the derivatives of any orders of them constitute Hilbertian bases in the space of weighted square integrable functions.     

On orthogonal polynomials obtained via polynomial mappings

Let (p_n)_n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k≥2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (q_n)_n and two polynomials π_k and θ_m, with degrees k and m (resp.), with 0≤m≤k−1, such that In this work we establish algebraic conditions for the existence of a polynomial mapping in the above sense. Under such conditions, when (p_n)_n is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS (p_n)_n in terms of the orthogonality measure for the OPS (q_n)_n. Some applications and examples are presented, recovering several known results in a unified way.     

On one set of orthogonal harmonic polynomials

A new basis of harmonic polynomials is given. Proposed polynomials are orthogonal on the unit sphere and each term of this basis consists of monomials not present in the others.

     

On generalized Stieltjes-Wigert and related orthogonal polynomials

The Charlier, Wall, and generalized Stieltjes-Wigert polynomials are characterized by a property involving the concept of kernel polynomials. This characterization leads to consideration of a certain functional equation satisfied by solutions of the associated Stieltjes moment problem. All distribution functions which satisfy this functional equation are found up to singular functions. This yields new distribution functions, both discrete and absolutely continuous, with respect to which generalized Stieltjes-Wigert polynomials are orthogonal.     

On differential properties for bivariate orthogonal polynomials

In this paper, we consider bivariate orthogonal polynomials associated with a quasi-definite moment functional which satisfies a Pearson-type partial differential equation. For these polynomials differential properties are obtained. In particular, we deduce some structure and orthogonality relations for the successive partial derivatives of the polynomials.      

On sieved orthogonal polynomials. VIII. Sieved associated Pollaczek polynomials

We study a general orthogonal polynomial set which includes the sieved associated ultraspherical and the sieved Pollaczek polynomials. This we get by letting q approach a root of unity in the recurrence relation and the generating functions of the associated q-ultraspherical and the Pollaczek polynomials. We find the weight functions with respect to which these polynomials are orthogonal and determine the asymptotic behavior of these polynomials on and off their interval of orthogonality.     

On the coefficients of integrated expansions of Bessel polynomials

A new formula expressing explicitly the integrals of Bessel polynomials of any degree and for any order in terms of the Bessel polynomials themselves is proved. Another new explicit formula relating the Bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established. An application of these formulae for solving ordinary differential equations with varying coefficients is discussed.     

On some orthogonal polynomials of interest in theoretical chemistry

Constructive methods are developed for a class of polynomials orthogonal on two symmetric intervals. An analysis is given of certain phenomena of instability in connection with nonlinear recursions. Special cases arising in the study of the diatomic linear chain are worked out explicitly. In one of these cases the associatedn-point Gauss-Christoffel quadrature formula has equal weights whenevern is even.To Germund Dahlquist on his 60th birthdayResearch supported in part by the National Science Foundation under grant MCS-7927158A1.