Approximation by polynomials in Sobolev spaces associated with classical moment functionals

Abstract

Let $$\textbf{u}$$ u be a moment functional associated with the Hermite, Laguerre, or Jacobi classical orthogonal polynomials. We study approximation by polynomials in $$H^r(\textbf{u})$$ H r ( u ) , the Sobolev space consisting of functions whose derivatives of consecutive orders up to r belong to the $$L^2$$ L 2 space associated with $$\textbf{u}$$ u . This requires the simultaneous approximation of a function f and its consecutive derivatives up to order $$N\leqslant r$$ N ⩽ r . We explicitly construct orthogonal polynomials that achieve such simultaneous approximation and provide error estimates in terms of $$E_n(f^{(r)})$$ E n ( f ( r ) ) , the error of best approximation of $$f^{(r)}$$ f ( r ) in $$L^{2}(\textbf{u})$$ L 2 ( u ) .