Zero location and asymptotic behavior for extremal polynomials with non-diagonal Sobolev norms

In this paper we are going to study the zero location and asymptotic behavior of extremal polynomials with respect to a non-diagonal Sobolev norm in the worst case, i.e., when the quadratic form is allowed to degenerate. 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.

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

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

[3]  A Kolmogorov-Szegö-Krein type condition for weighted Sobolev spaces , 2005 .

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

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

[6]  GENERALIZED WEIGHTED SOBOLEV SPACES AND APPLICATIONS TO SOBOLEV ORTHOGONAL POLYNOMIALS: A SURVEY , .

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

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

[9]  D. Lubinsky.  WEIERSTRASS' THEOREM IN THE TWENTIETH CENTURY: A SELECTION , 1995 .

[10]  E. Saff,et al.  Logarithmic Potentials with External Fields , 1997 .

[11]  F. Marcellán,et al.  Strong and PlancherelRotach Asymptotics of Non-diagonal LaguerreSobolev Orthogonal Polynomials , 2001 .

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

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

[14]  H. Dueñas,et al.  The holonomic equation of the Laguerre–Sobolev-type orthogonal polynomials: a non-diagonal case , 2011 .

[15]  A. Ronveaux,et al.  Sobolev-Type Orthogonal Polynomials: The Nondiagonal Case , 1995 .

[16]  P. Revesz.  Interpolation and Approximation , 2010 .

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

[18]  Sobolev Spaces with Respect to Measures in Curves and Zeros of Sobolev Orthogonal Polynomials , 2008, 0805.4761.

[19]  E. Cheney.  Introduction to approximation theory , 1966 .

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

[21]  José M. Rodríguez,et al.  A simple characterization of weighted Sobolev spaces with bounded multiplication operator , 2008, J. Approx. Theory.

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

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

[24]  Héctor Pijeira Cabrera,et al.  Asymptotic of extremal polynomials in the complex plane , 2005, J. Approx. Theory.

[25]  Arieh Iserles,et al.  Orthogonality and approximation in a Sobolev space , 1990 .

[26]  José M. Rodríguez,et al.  Weighted Sobolev Spaces on Curves , 2002, J. Approx. Theory.

[27]  Vilmos Totik,et al.  General Orthogonal Polynomials , 1992 .

[28]  Arieh Iserles,et al.  On polynomials orthogonal with respect to certain Sobolev inner products , 1991 .

[29]  L. Littlejohn,et al.  Ghost matrices and a characterization of symmetric Sobolev bilinear forms , 2009 .

[30]  The Multiplication Operator, Zero Location and Asymptotic for Non-diagonal Sobolev Norms , 2010 .

[31]  Francisco Marcellán,et al.  Asymptotic behaviour of Laguerre-Sobolev-type orthogonal polynomials. A nondiagonal case , 2010, J. Comput. Appl. Math..

[32]  Francisco Marcellán,et al.  Strong and Plancherel-Rotach Asymptotics of Non-diagonal Laguerre-Sobolev Orthogonal Polynomials , 2001, J. Approx. Theory.

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

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

[35]  R. Brown,et al.  Embeddings of weighted Sobolev spaces into spaces of continuous functions , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

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

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

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

[39]  József Szabados,et al.  Approximation with exponential weights in (−1, 1) , 2002 .

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

[41]  Allan Pinkus,et al.  Weierstrass and Approximation Theory , 2000 .