On the uniform approximation of functions of bounded variation by Lagrange interpolation polynomials with a matrix L-n((alpha n,beta n)) of Jacobi nodes

摘要

Let sequences {alpha(n)}(n=1)(infinity), {beta(n)}(n=1)(infinity) satisfy the relations alpha(n) is an element of R, beta(n) is an element of R, alpha(n) = o(root n/ln n), beta(n) = o(root n/ln n) as n -> infinity, and let [a, b] subset of (0, pi) and f is an element of C[a, b]. We redefine the function f as F on the interval [0, pi] by polygonal arcs in such a way that the function remains continuous and vanishes on a neighbourhood of the ends of the interval. Also let the function f and the pair of sequences {alpha(n)}(n=1)(infinity), {beta(n)}(n=1)(infinity) be connected by the equiconvergence condition. Then for the classical Lagrange-Jacobi interpolation processes L-n((alpha n,beta n)) (F, cos theta) to approximate f uniformly with respect to theta on [a, b] it is sufficient that f have bounded variation V-a(b) (f) < infinity on [a, b]. In particular, if the sequences {alpha(n)}(n=1)(infinity) and {beta(n)}(n=1)(infinity) are bounded, then for the classical Lagrange-Jacobi interpolation processes L-n((alpha n,beta n)) (F, cos theta) to approximate f uniformly with respect to. on [a, b] it is sufficient that the variation of f be bounded on [a, b], V-a(b) (f) < infinity.