Near-minimax complex approximation by four kinds of Chebyshev polynomial expansion

Abstract

Functions of the form w( z) F( z) with F analytic and w( z)=1, (z 2−1) 1 2 , (z + 1) 1 2 or (z−1) 1 2 are approximated in the ellipse ξ r:|z+(z 2−1) 1 2 |=r by w( z) p n ( z), where p n is a polynomial of degree n. Here p n is obtained by the expansion of F in Chebyshev polynomials of the first, second, third or fourth kinds, corresponding to the above four respective weight functions. Bounds are established and computed for the norms on ξ r of the corresponding projections, thus confirming that all resulting approximations are near-minimax within relative distances asymptotically proportional to 4π −2 ln n, and extending a known result (Geddes, 1978) for w( z)=1 and first kind Chebyshev polynomials.