Sponsor: NSF
This is ongoing work on constructive and computational aspects of orthogonal polynomials. Currently, the main focus is on orthogonal polynomials of Sobolev type, i.e., on polynomials orthogonal with respect to an inner product involving derivatives. Modified moment algorithms known from the theory of ordinary orthogonal polynomials are being extended, tested, and analyzed and so are Stieltjes-type algorithms. Special techniques are developed for Sobolev-type polynomials relative to an ordinary inner product superimposed by a one-point (atomic) inner product involving a derivative of fixed order. Other current research involves Stieltjes polynomials and related quadrature formulae, the computation of Turán quadrature rules based on the theory of s-orthogonality, and quadrature convergence of extended Lagrange interpolation.