Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials II

Abstract

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).