M.L. Rezola

Orthogonal polynomials on Sobolev spaces: old and new directions

Abstract During the last years, orthogonal polynomials on Sobolev spaces have attracted considerable attention. Algebraic properties, distribution of their zeros and Fourier expansions as well as their relevance in the analysis of spectral methods for partial differential equations provide a very large field to explore and to compare with the standard case. In this paper we present an introductory survey about the subject. The origin of the problems and their development show the interest and the promising future of this field.

Some properties of zeros of Sobolev-type orthogonal polynomials

Abstract 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.

Full length article: A new approach to the asymptotics of Sobolev type orthogonal polynomials

This paper deals with Mehler-Heine type asymptotic formulas for the so-called discrete Sobolev orthogonal polynomials whose continuous part is given by Laguerre and generalized Hermite measures. We use a new approach which allows to solve the problem when the discrete part contains an arbitrary (finite) number of mass points.

Discrete Laguerre–Sobolev expansions: A Cohen type inequality☆

Abstract C. Markett proved a Cohen type inequality for the classical Laguerre expansions in the appropriate weighted L p spaces. In this paper, we get a Cohen type inequality for the Fourier expansions in terms of discrete Laguerre–Sobolev orthonormal polynomials with an arbitrary (finite) number of mass points. So, we extend the result due to B.Xh. Fejzullahu and F. Marcellan.

Sobolev orthogonal polynomials: balance and asymptotics

Let µ0 and µ1 be measures supported on an unbounded interval and Sn, n the extremal varying Sobolev polynomial which minimizes

An explicit expression for the Kr functionals of interpolation between Lp spaces

When dealing with interpolation spaces by real methods one is lead to compute (or at least to estimate) the K-functional associated to the couple of interpolation spaces. This concept was first introduced by J. Peetre (see [8], [9]) and some efforts have been done to find explicit expressions of it for the case of Lebesgue spaces. It is well known that for the couple consisting of L1 and L8 on [0, 8) K is given by K (t; f, L1, L8) = ?0t f* where f* denotes the non increasing rearrangement of the function f. The aim of this paper is to answer a question raised by J. Peetre to the autors and to extend the results in [1] and [7] for the more general case of the Kr funcitionals between Lp spaces.

Connection formulas for general discrete Sobolev polynomials

In this paper the discrete Sobolev inner product { p , q } = ? p ( x ) q ( x ) d µ + ? i = 0 r M i p ( i ) ( c ) q ( i ) ( c ) is considered, where µ is a finite positive Borel measure supported on an infinite subset of the real line, c ? R and ?Mi ? 0,?i = 0, 1, ?, r.Connection formulas for the orthonormal polynomials associated with {., .} are obtained. As a consequence, for a wide class of measures µ, we give the Mehler-Heine asymptotics in the case of the point c is a hard edge of the support of µ. In particular, the case of a symmetric measure µ is analyzed. Finally, some examples are presented.

Orthogonal polynomials generated by a linear structure relation: Inverse problem

Abstract Let ( P n ) n and ( Q n ) n be two sequences of monic polynomials linked by a type structure relation such as Q n ( x ) + r n Q n − 1 ( x ) = P n ( x ) + s n P n − 1 ( x ) + t n P n − 2 ( x ) , where ( r n ) n , ( s n ) n and ( t n ) n are sequences of complex numbers. First, we state necessary and sufficient conditions on the parameters such that the above relation becomes non-degenerate when both sequences ( P n ) n and ( Q n ) n are orthogonal with respect to regular moment linear functionals u and v , respectively. Second, assuming that the above relation is non-degenerate and ( P n ) n is an orthogonal sequence, we obtain a characterization for the orthogonality of the sequence ( Q n ) n in terms of the coefficients of the polynomials Φ and Ψ which appear in the rational transformation (in the distributional sense) Φ u = Ψ v . Some illustrative examples of the developed theory are presented.