Constrained near-minimax approximation by weighted expansion and interpolation using Chebyshev polynomials of the second, third, and fourth kinds

Abstract

Well known results on near-minimax approximation using Chebyshev polynomials of the first kind are here extended to Chebyshev polynomials of the second, third, and fourth kinds. Specifically, polynomial approximations of degreen weighted by (1−x ²)^1/2, (1+x)^1/2 or (1−x)^1/2 are obtained as partial sums of weighted expansions in Chebyshev polynomials of the second, third, or fourth kinds, respectively, to a functionf continuous on [−1, 1] and constrained to vanish, respectively, at ±1, −1 or +1. In each case a formula for the norm of the resulting projection is determined and shown to be asymptotic to 4π⁻²logn +A +o(1), and this provides in each case and explicit bound on the relative closeness of a minimax approximation. The constantA that occurs for the second kind polynomial is markedly smaller, by about 0.27, than that for the third and fourth kind, while the latterA is identical to that for the first kind, where the projection norm is the classical Lebesgue constant λ _n . The results on the third and fourth kind polynomials are shown to relate very closely to previous work of P.V. Galkin and of L. Brutman.

Analogous approximations are also obtained by interpolation at zeros of second, third, or fourth kind polynomials of degreen+1, and the norms of resulting projections are obtained explicitly. These are all observed to be asymptotic to 2π⁻¹logn +B +o(1), and so near-minimax approximations are again obtained. The norms for first, third, and fourth kind projections appear to be converging to each other. However, for the second kind projection, we prove that the constantB is smaller by a quantity asymptotic to 2π⁻¹log2, based on a conjecture about the point of attainment of the supremum defining the projection norm, and we demonstrate that the projection itself is remarkably close to a minimal (weighted) interpolation projection.

All four kinds of Chebyshev polynomials possess a weighted minimax property, and, in consequence, all the eight approximations discussed here coincide with minimax approximations when the functionf is a suitably weighted polynomial of degreen+1.