Stieltjes polynomials and Lagrange interpolation

Bounds are proved for the Stieltjes polynomial E n+1 , and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials P n . This sharpens a known interlacing result of Szego. As a byproduct, bounds are obtained for the Geronimus polynomials G n . Applying these results, convergence theorems are proved for the Lagrange interpolation process with respect to the zeros of E n+1 , and for the extended Lagrange interpolation process with respect to the zeros of P n E n+1 in the uniform and weighted L P norms. The corresponding Lebesgue constants are of optimal order.

[1]  P. Nevai,et al.  Mean convergence of derivatives of Lagrange interpolation , 1991 .

[2]  D. Occorsio.  Una buona matrice di nodi di interpolazione , 1993 .

[3]  T. J. Rivlin,et al.  A family of Gauss-Kronrod quadrature formulae , 1988 .

[4]  Yuan Xu.  Mean convergence of generalized Jacobi series and interpolating polynomials, I , 1993 .

[5]  G. Szegö,et al.  Über gewisse orthogonale Polynome, die zu einer oszillierenden Belegungsfunktion gehören , 1935 .

[6]  Richard L. Wheeden,et al.  Two weight function norm inequalities for the Hardy-Littlewood maximal function and the Hilbert transform , 1976 .

[7]  Sven Ehrich.  Asymptotic properties of Stieltjes polynomials and Gauss-Kronrod quadrature formulae , 1995 .

[8]  Sotirios E. Notaris,et al.  An algebraic study of Gauss-Kronrod quadrature formulae for Jacobi weight functions , 1988 .

[9]  P. Nevai,et al.  Mean Convergence of Derivatives of Extended Lagrange Interpolation with Additional Nodes , 1993 .

[10]  Klaus-Jürgen Förster,et al.  On estimates for the weights in Gaussian quadrature in the ultraspherical case , 1990 .

[11]  G. Mastroianni.  Approximation of functions by extended Lagrange interpolation , 1994 .

[12]  Leonard Carlitz,et al.  A set of polynomials , 1940 .

[13]  P. Vértesi,et al.  Weighted Lp Error of Lagrange Interpolation , 1995 .

[14]  Franz Peherstorfer,et al.  On the asymptotic behaviour of functions of the second kind and Stieltjes polynomials and on the Gauss-Kronrod quadrature formulas , 1992 .

[15]  G. P. Névai,et al.  Mean convergence of Lagrange interpolation, II☆ , 1976 .

[16]  Giovanni Monegato,et al.  Stieltjes Polynomials and Related Quadrature Rules , 1982 .

[17]  T. J. Rivlin.  The Chebyshev polynomials , 1974 .

[18]  Giovanni Monegato,et al.  Orthogonal polynomials and numerical methods , 1992 .

[19]  Paul Nevai,et al.  Mean convergence of Lagrange interpolation. III , 1984 .

[20]  G. Mastroianni.  Uniform convergence of derivatives of Lagrange interpolation , 1992 .

[21]  Giovanni Monegato.  An overview of results and questions related to Kronrod schemes , 1979 .

[22]  L. Durand.  Nicholson-Type Integrals for Products of Gegenbauer Functions and Related Topics , 1975 .

[23]  Giovanni Monegato,et al.  Positivity of the weights of extended Gauss-Legendre quadrature rules , 1978 .

[24]  P. Vértesi,et al.  Pointwise simultaneous convergence of extended Lagrange interpolation with additional knots , 1992 .

[25]  G. Monegato.  A note on extended Gaussian quadrature rules , 1976 .

[26]  Giuseppe Mastroianni,et al.  Convergence of extended Lagrange interpolation , 1990 .

[27]  Quadrature sums involving p th powers of polynomials , 1987 .

[28]  Paul Neval,et al.  Hilbert transforms and Lagrange interpolation , 1990 .