On Hahn polynomial expansion of a continuous function of bounded variation

We consider the well-known method of least squares (cf., e.g., 1, p. 217]) on an equidistant grid with N + 1 nodes on the interval −1, 1]. We investigate the following problem: For which ratio N/n, do we have pointwise convergence of the least square operator LS^N_n : C−1,1]→P_n? To solve this problem we investigate the relation between the Jacobi polynomials P^α,β_k (cf., e.g., 2, p. 216]) and the Hahn polynomials Q_k (·; α, β, N) (cf., e.g., 2, p. 204]). In particular, we present the following result: Let f ∈ {g ∈ C¹−1, 1] : g′∈ BV −1, 1]} and let (N_n)_n be a sequence of natural numbers with n⁴/N_n → 0. Then the least square method LS^N_n f] converges for each x ∈ −1, 1]. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)