Sobolev orthogonal polynomials in two variables and second order partial differential equations

We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form

(∗)     

Strong asymptotics for Sobolev orthogonal polynomials in the complex plane

We obtain the strong asymptotics for the sequence of monic polynomials minimizing the norm

     

Rodrigues type formula for orthogonal polynomials on the unit ball

For a class of weight functions invariant under reflection groups on the unit ball, a family of orthogonal polynomials is defined via a Rodrigues type formula using the Dunkl operators. Their properties and their relation with several other bases are explored.

     

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.      

Zeros and logarithmic asymptotics of Sobolev orthogonal polynomials for exponential weights

We obtain the (contracted) weak zero asymptotics for orthogonal polynomials with respect to Sobolev inner products with exponential weights in the real semiaxis, of the form , with γ>0, which include as particular cases the counterparts of the so-called Freud (i.e., when φ has a polynomial growth at infinity) and Erdös (when φ grows faster than any polynomial at infinity) weights. In addition, the boundness of the distance of the zeros of these Sobolev orthogonal polynomials to the convex hull of the support and, as a consequence, a result on logarithmic asymptotics are derived.     

Classical orthogonal polynomials in two variables: a matrix approach

Classical orthogonal polynomials in two variables are defined as the orthogonal polynomials associated to a two-variable moment functional satisfying a matrix analogue of the Pearson differential equation. Furthermore, we characterize classical orthogonal polynomials in two variables as the polynomial solutions of a matrix second order partial differential equation. AMS subject classification 42C05, 33C50Partially supported by Ministerio de Ciencia y Tecnología (MCYT) of Spain and by the European Regional Development Fund (ERDF) through the grant BFM2001-3878-C02-02, Junta de Andalucía, G.I. FQM 0229 and INTAS Project 2000-272.     

A canonical family of multiple orthogonal polynomials for Nikishin systems

For any pair of compact intervals of the real line Δ₁, Δ₂, with Δ₁∩Δ₂=∅, we obtain two probability measures μ₁, τ₁, supported on Δ₁ and Δ₂ respectively, such that the Nikishin system N(μ₁,τ₁) has a sequence of monic multiple orthogonal polynomials which satisfy a four term recurrence relation with constant coefficients of period 2. The measures are obtained from the functions which give the ratio asymptotic of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ₁,σ₂) on Δ₁, Δ₂, such that a.e. on Δi, i=1,2. The role of μ₁, τ₁ is symmetric in the sense that the same construction is possible on Δ₂, Δ₁, with N(τ₁,μ₁).     

Relative asymptotics and Fourier series of orthogonal polynomials with a discrete Sobolev inner product

Let μ be a finite positive Borel measure supported in [−1,1] and introduce the discrete Sobolev-type inner product

where the mass points a_k belong to [−1,1], M_k,i0, i=0,…,N_k−1, and M_k,Nk>0. In this paper, we study the asymptotics of the Sobolev orthogonal polynomials by comparison with the orthogonal polynomials with respect to the measure μ and we prove that they have the same asymptotic behaviour. We also study the pointwise convergence of the Fourier series associated to this inner product provided that μ is the Jacobi measure. We generalize the work done by F. Marcellán and W. Van Assche where they studied the asymptotics for only one mass point in [−1,1]. The same problem with a finite number of mass points off [−1,1] was solved by G. López, F. Marcellán and W. Van Assche in a more general setting: they consider the constants M_k,i to be complex numbers. As regards the Fourier series, we continue the results achieved by F. Marcellán, B. Osilenker and I.A. Rocha for the Jacobi measure and mass points in .     

On five-diagonal Toeplitz matrices and orthogonal polynomials on the unit circle

This paper deals with modifications of the Lebesgue moment functional by trigonometric polynomials of degree 2 and their associated orthogonal polynomials on the unit circle. We use techniques of five-diagonal matrix factorization and matrix polynomials to study the existence of such orthogonal polynomials.Dedicated to Prof. Luigi Gatteschi on his 70th birthdayThis research was partially supported by Diputación General de Aragón under grant P CB-12/91.     

Generalized Weighted Sobolev Spaces and Applications to Sobolev Orthogonal Polynomials I

In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.     

On the linear functionals associated to linearly related sequences of orthogonal polynomials

An inverse problem is solved, by stating that the regular linear functionals u and v associated to linearly related sequences of monic orthogonal polynomials n(Pn) and n(Qn), respectively, in the sense

     

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     

Asymptotics and Zeros of Sobolev Orthogonal Polynomials on Unbounded Supports

In this paper we present a survey about analytic properties of polynomials orthogonal with respect to a weighted Sobolev inner product such that the vector of measures has an unbounded support. In particular, we focus on the asymptotic behaviour of such polynomials as well as in the distribution of their zeros. Some open problems as well as some directions for future research are formulated.Research of Juan José Moreno Balcázar was partially supported by Ministerio de Educación y Ciencia of Spain under grant MTM2005-08648-C02-01 and Junta de Andalucía (FQM 229 and FQM 481).     

Differential operators having Sobolev type Laguerre polynomials as eigenfunctions

We consider the polynomials orthogonal with respect to the Sobolev type inner product

where and is a nonnegative integer. It is the purpose of this paper to show that these polynomials are eigenfunctions of a class of linear differential operators containing one that is of finite order if is a nonnegative integer and

     

Second order partial differential equations for gradients of orthogonal polynomials in two variables

In this work, we introduce the classical orthogonal polynomials in two variables as the solutions of a matrix second order partial differential equation involving matrix polynomial coefficients, the usual gradient operator, and the divergence operator. Here we show that the successive gradients of these polynomials also satisfy a matrix second order partial differential equation closely related to the first one.     

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.     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.