Weighted Sobolev spaces: Markov-type inequalities and duality (original) (raw)

Markov-type inequalities and duality in weighted Sobolev spaces

The aim of this paper is to provide Markov-type inequalities in the setting of weighted Sobolev spaces when the considered weights are generalized classical weights. Also, as results of independent interest, some basic facts about Sobolev spaces with respect to certain vector measures are stated.

Some Markov-Bernstein type inequalities and certain class of Sobolev polynomials

Let (μ0, μ1) be a vector of non-negative measures on the real line, with μ0 not identically zero, finite moments of all orders, compact or non compact supports, and at least one of them having an infinite number of points on its support. We show that for any linear operator T on the space of polynomials with complex coefficients and any integer n ≥ 0, there is a constant γn(T ) ≥ 0, such that T p S ≤ γn(T ) p S , for any polynomial p of degree ≤ n, where γn(T ) is independent of p, and

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.

A Kolmogorov-Szego-Krein type condition for weighted Sobolev spaces

Indiana University Mathematics Journal, 2005

§0. Introduction and main results Let p ∈ [1, +∞), k ∈ N, and let µ = (µ 0 , µ 1 ,. .. , µ k) be a (k + 1)-tuple of positive finite Borel measures on the unit circle T = {z : |z| = 1} in the complex plane. Consider the continuous mapping Π : C k (T) → k j=0 C(T), given by Πf = f, f ,. .. , f (k) , where f (z) = df dz (all spaces of functions that we consider are complex-valued). Note that df dz (e iθ) = −ie −iθ d dθ f (e iθ). Definition. The abstract Sobolev space W k,p (µ) = W k,p (µ 0 ,. .. , µ k) is the closure of ΠC k (T) in the space k j=0 L p (T, µ j). We refer to [17] for the classical theory of Sobolev spaces in domains of R n. We refer to [13, 18, 5, 12] for the theory of weighted Sobolev spaces in domains of R n ; in [11, 14], one can find applications of this topic to partial differential equations. We consider in W k,p (µ) the usual norm f k,p,µ = k j=0 f j p p,µ j 1/p , f = (f 0 ,. .. , f k). Each function f in C k (T) has its image Πf in W k,p (µ), and these images are dense in W k,p (µ). In many cases, an element g = (g 0 ,. .. , g k) in W k,p (µ) is completely determined by its first component g 0 , so that W k,p (µ) can be identified with a certain space of measurable functions g 0 , and the components g 1 ,. .. , g k can be thought of as a kind of generalized derivatives of g 0. In general, however, elements of W k,p (µ) cannot be identified with scalar functions on T. This setting of abstract Sobolev spaces is the most natural for us. (See [2], [22]-[26] in order to know when W k,p (µ) is a space of functions.) This space plays a central role in the theory of orthogonal polynomials with respect to Sobolev inner products (see [2], [15], [16] and [23]; in [2] and [16], the authors consider measures supported in compact sets in the complex plane). In fact, if the multiplication operator (M f)(z) = zf (z) is bounded in W k,2 (µ), then every 1 The research of the first author was partially supported by two grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. 2 The research of the second author was partially supported by the Ramón y Cajal Programme by the Ministry of Science and Technology of Spain.

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.