Density of polynomials in the L 2 space on the real and the imaginary axes and in a Sobolev space (original) (raw)

Density of polynomials in some L2 spaces on radial rays in the complex plane

Linear Algebra and its Applications, 2011

In this manuscript we study necessary and sufficient conditions for the density of the linear space of matrix polynomials in a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays of the complex plane. The connection with a completely indeterminate Hamburger matrix moment problem is stated. Vector valued functions associated in a natural way with a function defined in the union of the radial rays are used. Thus, our first aim is the construction of a linear space of square integrable functions with respect to a matrix of measures supported on a set of radial rays and a positive semi-definite matrix acting on the discrete part of the corresponding inner product. An isometric transformation which allows to reduce the problem of density to the case of the real line is introduced. Finally, some examples of such spaces are shown and its completeness is studied in detail.

On a class of Sobolev scalar products in the polynomials

Journal of Approximation Theory, 2003

This paper discusses Sobolev orthogonal polynomials for a class of scalar products that contains the sequentially dominated products introduced by Lagomasino and Pijeira. We prove asymptotics for Markov type functions associated to the Sobolev scalar product and an extension of Widom's Theorem on the location of the zeroes of the orthogonal polynomials. In the case of measures supported in the real line, we obtain results related to the determinacy of the Sobolev moment problem and the completeness of the polynomials in a suitably defined weighted Sobolev space.

Orthogonal polynomials and approximation in Sobolev spaces

Journal of Computational and Applied Mathematics, 1993

Everitt, W.N., L.L. Littlejohn and SC. Williams, Orthogonal polynomials and approximation in Sobolev spaces, Journal of Computational and Applied Mathematics 48 (1993) 69-90. This paper discusses the density of polynomials in Sobolev-type function spaces defined on the compact interval [ -1, l] of the real line W. The problems considered are motivated by consideration of the spectral representation of certain Jacobi-type orthogonal polynomials.

Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials

Acta Applicandae Mathematicae, 2008

In this paper we obtain some practical criteria to bound the multiplication operator in Sobolev spaces with respect to measures in curves. As a consequence of these results, we characterize the weighted Sobolev spaces with bounded multiplication operator, for a large class of weights. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials. We also obtain some non-trivial results about these Sobolev spaces with respect to measures; in particular, we prove a main result in the theory: they are Banach spaces.