Weighted Sobolev Spaces on Curves

In this paper we present a definition of weighted Sobolev spaces on curves and find general conditions under which the spaces are complete for non-closed compact curves. We also prove the density of the polynomials in these spaces and, finally, we find conditions under which the multiplication operator is bounded in the space of polynomials.

[1]  José M. Rodríguez,et al.  The Multiplication Operator in Sobolev Spaces with Respect to Measures , 2001, J. Approx. Theory.

[2]  A. Martínez-Finkelshtein.  Bernstein—Szegő's Theorem for Sobolev Orthogonal Polynomials , 2000 .

[3]  J. García-cuerva,et al.  Weighted norm inequalities and related topics , 1985 .

[4]  José M. Rodríguez,et al.  Generalized Weighted Sobolev Spaces and Applications to Sobolev Orthogonal Polynomials I , 2004 .

[5]  F. Marcellán,et al.  Asymptotic Behavior of Sobolev-Type Orthogonal Polynomials on a Rectifiable Jordan Curve or Arc , 2002 .

[6]  Asymptotic behavior of Sobolev-type orthogonal polynomials on the unit circle , 1999 .

[7]  Weierstrass' Theorem in Weighted Sobolev Spaces , 2001 .

[8]  José M. Rodríguez.  Weierstrass' Theorem in Weighted Sobolev Spaces , 2001, J. Approx. Theory.

[9]  The Multiplication Operator in Sobolev Spaces with Respect to Measures , 2001 .

[10]  Sobolev orthogonal polynomials in the complex plane , .

[11]  VenancioAlvarez.  GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS II , 2002 .

[12]  Lance L. Littlejohn,et al.  Orthogonal polynomials and approximation in Sobolev spaces , 1993 .

[13]  WILLIAM H. ROWAN,et al.  Uniform Algebras , 2000 .

[14]  B. Muckenhoupt.  Hardy's inequality with weights , 1972 .

[15]  T. Kilpeläinen,et al.  Weighted Sobolev spaces and capacity. , 1994 .

[16]  H. Triebel.  Interpolation Theory, Function Spaces, Differential Operators , 1978 .

[17]  A. Kufner,et al.  Some applications of weighted Sobolev spaces , 1987 .

[18]  H. P. Cabrera,et al.  Zero Location and nth Root Asymptotics of Sobolev Orthogonal Polynomials , 1999 .

[19]  José M. Rodríguez,et al.  Approximation by polynomials and smooth functions in Sobolev spaces with respect to measures , 2003, J. Approx. Theory.

[20]  J. Heinonen,et al.  Nonlinear Potential Theory of Degenerate Elliptic Equations , 1993 .

[21]  Seng-Kee Chua,et al.  On Weighted Sobolev Spaces , 1996, Canadian Journal of Mathematics.

[22]  Bohumír Opic,et al.  How to define reasonably weighted Sobolev spaces , 1984 .

[23]  B. Muckenhoupt,et al.  Weighted norm inequalities for the Hardy maximal function , 1972 .

[24]  G. Lo.  Zero Location and n th Root Asymptotics of Sobolev Orthogonal Polynomials , 1999 .

[25]  Sobolev orthogonal polynomial in the complex plane , 2001 .