Approximation by polynomials in Sobolev spaces associated with classical moment functionals

Abstract

Let u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in Hr(u), the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the L2 space associated with u. This requires the simultaneous approximation of a function f and its consecutive derivatives up to order N⩽r. We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of En(f(r)), the error of best approximation of f(r) in L2(u).