Positive convergent approximation operators associated with orthogonal polynomials for weights on the whole real line* 1 (original) (raw)

Polynomial approximation with Pollaczeck-Laguerre weights on the real semiaxis. A survey

ETNA - Electronic Transactions on Numerical Analysis, 2018

This paper summarizes recent results on weighted polynomial approximations for functions defined on the real semiaxis. The function may grow exponentially both at 0 and at +∞. We discuss orthogonal polynomials, polynomial inequalities, function spaces with new moduli of smoothness, estimates for the best approximation, Gaussian rules, and Lagrange interpolation with respect to the weight w(x) = x γ e −x −α −x β (α > 0, β > 1, γ ≥ 0).

A Survey of Weighted Polynomial Approximation with Exponential Weights

2007

Let W : R → (0,1] be continuous. Bernstein’s approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → k fW k L1(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950’s. Quantitative forms of the problem were actively investigated starting from the 1960’s. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov–Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0, ∞). Nor do we discuss Saff’s weighted approximation problem, nor the asymptotics of orthogonal polynomials.

A survey of weighted polynomial approximation with

2007

Let W : R → (0, 1] be continuous. Bernstein's approximation problem, posed in 1924, deals with approximation by polynomials in the weighted uniform norm f → f W L∞(R). The qualitative form of this problem was solved by Achieser, Mergelyan, and Pollard, in the 1950's. Quantitative forms of the problem were actively investigated starting from the 1960's. We survey old and recent aspects of this topic, including the Bernstein problem, weighted Jackson and Bernstein Theorems, Markov-Bernstein and Nikolskii inequalities, orthogonal expansions and Lagrange interpolation. We present the main ideas used in many of the proofs, and different techniques of proof, though not the full proofs. The class of weights we consider is typically even, and supported on the whole real line, so we exclude Laguerre type weights on [0, ∞). Nor do we discuss Saff's weighted approximation problem, nor the asymptotics of orthogonal polynomials. MSC: 41A10, 41A17, 41A25, 42C10 Keywords: Weighted approximation, polynomial approximation, Jackson-Bernstein theorems Contents 1 Bernstein's Approximation Problem 2 Some Ideas for the Resolution of Bernstein's Problem 3 Weighted Jackson and Bernstein Theorems 4 Methods for Proving Weighted Jackson Theorems 19 4.

An identity for a general class of approximation operators

Journal of Approximation Theory, 2006

We prove an identity for basis functions of a general family of positive linear operators. It covers as special cases the Bernstein, Szász-Mirakjan and Baskakov operators. A corollary of our result can be considered a pointwise orthogonality relation. The Bernstein case is the univariate case of a remarkable identity which recently was presented by Jetter and Stöckler. As an application we give a representation of a restricted dual basis and define a class of quasi-interpolants.

Approximation Results for Orthogonal Polynomials in Sobolev Spaces.

1982

Abstract: We analyze the approximation properties of some interpolation operators and some $ L_\ omega^ 2$-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function omega(x1,ldots,xd)\ omega ({x_1},\ ldots,{x_d}) omega(x1,ldots,xd), $ d\ geqslant 1$. The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces $ H_\ omega^ s $.