Extended Lagrange interpolation in weighted uniform norm (original) (raw)

Extended Lagrange interpolation on the real line

Journal of Computational and Applied Mathematics, 2014

Let {p m (w α )} m be the sequence of the polynomials orthonormal w.r.t. the Sonin-Markov weight w α (x) = e −x 2 |x| α . The authors study extended Lagrange interpolation processes essentially based on the zeros of p m (w α )p m+1 (w α ), determining the conditions under which the Lebesgue constants, in some weighted uniform spaces, are optimal.

Mean convergence of an extended Lagrange interpolation process on [0,+∞)

Acta Mathematica Hungarica, 2014

In this paper we study some extended Lagrange interpolation processes based on the zeros of the Generalized Laguerre polynomials. We give necessary and sufficient conditions such that the convergence of these processes, in suitable L p weighted spaces on the real semiaxis, is assured for 1 < p < +∞.

Weighted Hermite–Fejér interpolation on Laguerre nodes

Acta Mathematica Hungarica - ACTA MATH HUNG, 2003

We give a weighted Hermite–Fejr type interpolation process on the half line. On suitable Laguerre nodes it converges for continuous functions which fulfil a certain not too fast growing property at zero and infinity.

Interpolation and the Laguerre-Pólya class

A long standing open problem, known as the Karlin-Laguerre problem, in the study of the distribution of real zeros of a polynomial is to characterize all real sequences T={γ_k}_{k=0}^∞ such that they satisfy the property Z_c (T[p(x)])≤Z_c (p(x)), where Z_c(p(x)) denotes the number of non-real zeros of the real polynomial p(x)=∑_{k=0}^{n} a_k x_k and T(p(x))=∑_{k=0}^{n}γ_k a_k x_k. The main result of this paper shows that under a mild growth restriction, an entire function of exponential type f(z) for which the sequence T={f(k)}_{k=0}^{∞} satisfies the above condition must have only real zeros. The paper concludes with some applications to the Riemann hypothesis.

Approximation by max-product Lagrange interpolation operators

2011

The aim of this note is to associate to the Lagrange interpolatory polynomials on various systems of nodes (including the equidistant and the Jacobi nodes), continuous piecewise rational interpolatory operators of the so-called max-product kind, uniformly convergent to the function f , with Jackson-type rates of approximation. Mathematics Subject Classification (2010): 41A05, 41A25, 41A35.

Lagrange interpolation on the semiaxis. A survey

Journal of Interpolation and Approximation in Scientific Computing, 2012

In this brief survey are collected some recent results about optimal interpolation processes of Lagrange type based on the zeros of generalized Laguerre polynomials, i.e. the sequence of orthogonal polynomials {p m (w α)} m where w α (x) = e −x β x α. A new extended Lagrange process having optimal Lebesgue constants is also introduced.