Generalized Alternating Polynomials, some Properties and Numerical Applications

In the present paper we introduce the generalized alternatingpolynomials , the coefficients of which are defined together with the parameter W_n by the linear system where T_n(x) = cos (n arc cos x), is a set of n+2 distinct points in the interval [–1, 1], and fis a continuous function on [–1, 1], For the set of nodesx_k = cos [k/(n+1)] the w_n-polynomials coincide with the polynomialsof equiamplitude alternation introduced by de La Vall?e-Poussinand discussed in the literature earlier (Eterman, Malozemov,Meinardus, Cheney & Rivlin, Phillips & Taylor, Brutman,and others). It is shown that the generalized alternating polynomials arerelated to the polynomials of interpolation through the Lanczoseconomization process. Some approximation properties of w_n -polynomialsand W_n -parameters are studied. The application of w_n -polynomialsto function approximation and to the estimate of remainder termsfor quadrature formulas is discussed.     

Interpolation by Polynomials in z and z-1 on an Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zⁿ=Rⁿeⁱⁿ (0<<2/n) and the n roots of zⁿ=rⁿ.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations.