Weighted Sobolev Spaces on Curves

Álvarez, Venancio; Pestana, Domingo; Rodríguez, José M.; Romera, Elena

doi:10.1006/jath.2002.3709

Cited by 19 publications

(51 citation statements)

References 16 publications

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“…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…”

Section: Introductionmentioning

confidence: 99%

Measurable diagonalization of positive definite matrices

Quintana

Rodríguez

2014

Journal of Approximation Theory

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In this paper we show that any positive definite matrix V with measurable entries can be written as V = U ΛU * , where the matrix Λ is diagonal, the matrix U is unitary, and the entries of U and Λ are measurable functions (U * denotes the transpose conjugate of U ).This result allows to obtain results about the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which products of derivatives of different order appear. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials.MSC: 41A10; 46E35; 46G10; 47A56

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Section: Introductionmentioning

confidence: 99%

Measurable diagonalization of positive definite matrices

Quintana

Rodríguez

2014

Journal of Approximation Theory

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“…The papers [1], [2], [4], [11], [20] and [22] deal with Sobolev spaces on curves and more general subsets of the complex plane.…”

Section: Introductionmentioning

confidence: 99%

“…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…”

Section: Introductionmentioning

confidence: 99%

“…(see Theorem 2.2 below, which is [1,Theorem 8.1] in the case N = 1). The rest of the papers mention several conditions which guarantee the equivalence of norms in Sobolev spaces, and consequently, the boundedness of M .…”

Section: Introductionmentioning

confidence: 99%

“…Let {q n } n≥0 be a sequence of extremal polynomials with respect to (1). Then the zeros of the polynomials in {q n } n≥0 are uniformly bounded in the complex plane.…”

Section: Introductionmentioning

confidence: 99%

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

2009

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Abstract. In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a generalized non-diagonal Sobolev norm in which the product of the function and its derivative appears. The orthogonal polynomials with respect to this Sobolev norm are a particular case of those extremal polynomials. The multiplication operator by the independent variable is the main tool in order to obtain our results.

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Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

Rodríguez

Álvarez

Romera

et al. 2002

Approx. Theory & its Appl.

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Now, let us state our main results here. We refer to the definitions in the next section. Let us present first the theorem about completeness, which is proved in [RARP].Remark. The condition (Ω, µ) ∈ C is not very restrictive (see Definition 16 and its remarks in Section 3).We cannot obtain such a general result for the density of C ∞ (R); we need some additional hypotheses on the measures. The measures we are dealing with have been divided in five types and we have results which allow "gluing" these measures (see theorems 4.2, 4.3 and 4.4 below).These are our main results on density. Theorem 4.1. Let us consider 1 ≤ p < ∞ and µ = (µ 0 , . . . , µ k ) a p-admissible vectorial measure in [a, b]. If µ is a measure of type 1, 2, 3, 4 or 5, then C ∞ c (R) is dense in the Sobolev space W k,p ([a, b], µ). Theorems A and 4.1 have the following important consequence. Corollary 4.1. Let us consider 1 ≤ p < ∞ and µ = (µ 0 , . . . , µ k ) a p-admissible vectorial measure in [a, b]. If µ is a measure of type 1, 2, 3, 4 or 5, then W k,p ([a, b], µ) is the closure of the polynomials in the norm of W k,p ([a, b], µ).Observe that we cannot expectIn the following theorems we present density results for measures which can be obtained by "gluing" simpler ones (for example, gluing measures of types 1 to 5). Theorem 4.2. Let us consider 1 ≤ p < ∞ and −∞ ≤ a < b < c < d ≤ ∞. Let µ = (µ 0 , . . . , µ k ) be a p-admissible vectorial measure in [a, d], and assume that there exists an interval I ⊆ [b, c] with (I, µ) ∈ C 0 and µ j (I) < ∞ for 0 ≤ j ≤ k. Then C ∞ (R) is dense in W k,p ([a, d], µ) if and only if C ∞ (R) is dense in W k,p ([a, c], µ) and W k,p ([b, d], µ). Remark. If a, d ∈ R, we can write in Theorem 4.2 C ∞ c (R) instead of C ∞ (R).

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Weighted Sobolev Spaces on Curves

Cited by 19 publications

References 16 publications

Measurable diagonalization of positive definite matrices

Measurable diagonalization of positive definite matrices

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

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

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