Weierstrass' theorem in weighted Sobolev spaces with k derivatives

36 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.MR#: MR2382639 (2008m:41022)Zbl#: Zbl 1177.41007We characterize the set of functions which can be approximated by smooth functions and by polynomials with the norm $$\ \ ^{(j)}\ {W^{k,\infty}_w}=\sum_ {j=0}^k\ {L^{\infty}_{w_j}},$$ for a wide range of (possibly unbounded) vector weights $w=(w_0,\dots, w_k)$. We allow a great deal of independence among the weights $w=(w_0,\dots, w_k)$.Research of the first, thord and fourth authors partially supported by grants from DGI (MTM 2006-13000-C03-02, MTM 2003-11976 and MTM 2006-26627-E), Spain.Publicad

A unified framework for parabolic equations with mixed boundary conditions and diffusion on interfaces

In this paper we consider scalar parabolic equations in a general non-smooth setting with emphasis on mixed interface and boundary conditions. In particular, we allow for dynamics and diffusion on a Lipschitz interface and on the boundary, where diffusion coefficients are only assumed to be bounded, measurable and positive semidefinite. In the bulk, we additionally take into account diffusion coefficients which may degenerate towards a Lipschitz surface. For this problem class, we introduce a unified functional analytic framework based on sesquilinear forms and show maximal regularity for the corresponding abstract Cauchy problem.Comment: 27 pages, 4 figure

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials: a survey

6 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.MR#: MR2219917 (2006k:42051)Zbl#: Zbl 1146.42005In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.Research by first author (J.M.R.) was partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain. Research by second author (V.A.) was partially supported by grants from MCYT (MTM 2004-00078) and Junta de Andalucía (FQM-210), Spain. Research by third author (E.R.) was partially supported by a grant from DGI (BFM 2003-06335-C03-02), Spain. Research by fourth author (D.P.) was partially supported by grants from DGI (BFM 2003-06335-C03-02 and BFM 2003-04870), Spain.Publicad

Generalized weighted Sobolev spaces and applications to Sobolev orthogonal polynomials, I

36 pages, no figures.-- MSC2000 codes: 41A10, 46E35, 46G10.-- Part II of this paper published in: Approx. Theory Appl. 18(2): 1-32 (2002), available at: http://e-archivo.uc3m.es/handle/10016/6483MR#: MR2047389 (2005k:42062)Zbl#: Zbl 1081.42024In this paper we present a definition of Sobolev spaces with respect to general measures, prove some useful technical results, some of them generalizations of classical results with Lebesgue measure and find general conditions under which these spaces are complete. These results have important consequences in approximation theory. We also find conditions under which the evaluation operator is bounded.Research by first (J.M.R.), third (E.R.) and fourth (D.P.) authors was partially supported by a grant from DGI (BFM 2000-0206-C04-01), Spain.Publicad