Sobolev-type Orthogonal Polynomials Research Articles

Some properties of zeros of Sobolev-type orthogonal polynomials

For polynomials orthogonal with respect to a discrete Sobolev product, we prove that, for each n, Q n has at least n − m zeros on the convex hull of the support of the measure, where m denotes the number of terms in the discrete part. Interlacing properties of zeros are also described.

Sobolev-Type Orthogonal Polynomials: The Nondiagonal Case

Sobolev type orthogonal polynomials have been the object of increasing interest in the last few years. In this paper we introduce a generalization of the usual Sobolev-type inner product and we compare it with the strict diagonal case. Zeros and asymptotic properties of these kinds of polynomial sequence are studied.

Symmetrical Orthogonal Polynomials for Sobolev-Type Inner Products

In this paper, families of symmetric orthogonal polynomials (Qn) with respect to the Sobolev-type inner product, 〈f, g〉=∫Ifgdμ+Σrj=0Mjƒ(j)(0) g(j)(0) where I is a symmetric interval and μ is a symmetric positive Borel measure with infinite support on I and whose moments are all finite, are considered. If Q2n(x)=Un(x2) and Q2n+1(x)=xVn(x2), we deduce that Un and Vn are Sobolev-type orthogonal polynomials and, in several particular cases, standard orthogonal polynomials. We study the zeros of Qn showing that, in some cases, Qn has two complex conjugate zeros; moreover a partial result about separation of the zeros is given. We also discuss the symmetrization problem for this kind of inner products. Finally, some Sobolev-type inner products with two symmetric mass points are considered.

On higher order Padé-type approximants with some prescribed coefficients in the numerator

In this work, we consider the construction of higher order rational approximants to a formal power series, with some prescribed coefficients in their numerators, precisely those of the higher order powers. The denominators of such approximants are related to the so-called Sobolev-type orthogonal polynomials. The elementary properties of these orthogonal polynomials are studied in the regular case.