The Polya algorithm in sequence spaces

AbstractIn this paper we consider the problem of best approximation in ℓp(N), 1<p⩽∞. If hp, 1<p<∞ denotes the best ℓp-approximation of the element h∈ℓ1(N) from a finite-dimensional affine subspace K of ℓ1(N), h∉K, then limp→∞hp=h∞*, where h∞* is a best uniform approximation of h from K, the so-called strict uniform approximation. Our aim is to give a complete description of the rate of convergence of ∥hp-h∞*∥ as p→∞ by proving that there are constants L1,L2>0 and 0⩽a⩽1 such that L1ap⩽p∥hp-h∞*∥⩽L2ap,for p large enough