Explicit representations of the norms of the Laguerre-Sobolev and Jacobi-Sobolev polynomials

Numerical Algorithms, vol. 102, pp. 1635–1659

Abstract

Abstract This paper deals with discrete Sobolev orthogonal polynomials with respect to inner products built upon the classical Laguerre and Jacobi measures on the intervals $$ [0,\infty ) $$ [ 0 , ∞ ) and $$ [-1,1] $$ [ - 1 , 1 ] , respectively. In addition, they are equipped with point masses at a finite endpoint of the interval involving the underlying functions and their derivatives of first or higher order. One of the intrinsic features of these polynomials are their $$ L^2 $$ L 2 -norms in the corresponding inner product spaces. Their knowledge is essential to orthonormalize the polynomials and thus indispensable to treat the corresponding Fourier-Sobolev series and other topics, notably in approximation theory, spectral theory or mathematical physics. Proceeding from an appropriate representation of the Sobolev polynomials which reflect the influence of the point masses, we explicitly establish their squared norm in an efficient form. In each case, the value differs from the familiar squared norm of the Laguerre or Jacobi polynomials by a factor which itself is a product of two essentially identical terms. Surprisingly, each of these factors turns out to be the quotient of the leading coefficients of the Sobolev polynomial and its classical counterpart. Obviously, our results enable to determine the asymptotic behavior of the norms of the orthogonal polynomials considered for large n .

Authors 1

1. Clemens Markett corresponding Aachen Lehrstuhl A für Mathematik

   RWTH Aachen University

   Affiliation as printed

   Lehrstuhl A für Mathematik, RWTH Aachen, Templergraben 55, 52062, Aachen, Germany