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

Abstract

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.

Weighted Sobolev spaces: Markov-type inequalities and duality

“…We want to remark that the proof of Theorem 4.1 is very long and technical: the paper [36] is mainly devoted to prove a weak version of Theorem 4.1, and using this version, [39] provides a very technical proof of the general case. Note that it took 6 years to prove this natural property of Sobolev spaces with respect to measures.…”

“…These Sobolev spaces appear in a natural way and are a very useful tool when we study the asymptotic behavior of Sobolev orthogonal polynomials (see [7,[19][20][21][33][34][35]37,39]). In particular, the completeness of these spaces has been shown very useful; it is a remarkable fact that it took 6 years to prove this natural property (see [2,36,37,39]).…”

“…In particular, the completeness of these spaces has been shown very useful; it is a remarkable fact that it took 6 years to prove this natural property (see [2,36,37,39]). …”

Measurable diagonalization of positive definite matrices

“…In [1,3,[13][14][15][16][17] there are some answers to the question stated in [8] about some conditions for M to be bounded: the more general result on this topic is [1, Theorem 8.1] which characterizes in a simple way (in terms of equivalent norms in Sobolev spaces) the boundedness of M for the classical diagonal case…”

The Multiplication Operator, Zero Location and Asymptotic for Non-diagonal Sobolev Norms

“…In [1], [28], [30], [31] and [32], there are some answers to the question stated in [21] about some conditions for M to be bounded: the more general result on this topic is [1, Theorem 8.1] which characterizes in a simple way (in terms of equivalent norms in Sobolev spaces) the boundedness of M for the classical "diagonal" case…”