Laurent-Padé approximants to four kinds of Chebyshev polynomial expansions. Part I. Maehly type approximants

Abstract

Laurent Pade-Chebyshev rational approximants,A m (z,z −1)/B n (z, z −1), whose Laurent series expansions match that of a given functionf(z,z −1) up to as high a degree inz, z −1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Pade approximants of the same form, which matched expansions betweenf(z,z −1)B n (z, z −1)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Pade-Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Pade-Chebyshev coefficients are similar to that for a traditional Pade approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Pade-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Pade-Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper [7] Pade-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.