Some properties of the Chebyshev method

Abstract

Several properties of the Chebyshev method of summability, defined by G. G. Bilodeau, are investigated. Specifically, it is shown that the Chebyshev method is translative and is a Gronwall method. It is shown that the de Vallee Poussin method is stronger than the Chebyshev method, and that the Chebyshev method is not stronger than the (C, 1) method. The final result shows that the Chebyshev method exhibits the Gibbs phenomenon. Let Σ( — iyut be an alternating series with partial sums sn = ΣΓ=o( —l)*w< Define a sequence of polynomials {PJt)} by PJt) = Σ*=o«•***, P*(l) = 1, n = 0, 1, 2, •••. The series £(-1)%, will be called summable (PJ to the value s if lim σ(Pn) = 8, where σ(Pn) — ΣLo #«&£*• Bilodeau [1] considered the following question. What are sufficient conditions on Pn for σ(Pn) to speed up the rate of convergence of a convergent sequence {sn}Ί For sequences {un} which are moment sequences, i.e., un has the representation un = \tnda(t), Jo where a(t)eBV[0fϊ\, he obtains the estimate \σ(Pn) — s\/\rn ^