Chebyshev–Laurent Polynomials and Weighted Approximation

Let (a, b) ⊂ (0, ∞) and for any positive integer n, let S_n be the Chebyshev space in [a, b] defined by S_n := span { x⁻ⁿ /^2+k , k = 0,…, n }. The unique (up to a constant factor) function τ_n ∈ S_n , which satisfies the orthogonality relation ∫ a b τ n ( x ) q ( x ) ( x ( b − x ) ( x − a ) ) − 1 / 2 d x = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072126/15070696-a94f-4da5-a117-8ee069a896b3/content/eq1.tif"/> for any q ∈ S _{n – 1}, is said to be the orthogonal Chebyshev S_n -polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S_n -polynomials and to demonstrate their importance to the problem of approximation by S_n -polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S_n is discussed. It is shown also that τ_n obeys an extremal property in L_q , 1 ≤ q ≤ ∞. Natural analogues of some inequalities for algebraic polynomials, which we expect to hold for the S_n -polynomials, are conjectured.