Histospline projections on a uniform partition

Abstract

We present a unified treatment of the periodic histospline projection of a function f on a uniform partition. We consider a given real number v ϵ [0, 1] and obtain existence and uniqueness results for the n-degree periodic spline s determined by the values { ∫ x i+vh x i+(v+1)h s(x)dx=0 N−1 . For a function f ∈ C p n + 1 [ a, b] and a spline determined by the conditions ∫ x i+vh x i+(v+1)h s(x)dx= ∫ x i+vh x i+(v+1)h f(x)dx(i=0,…,N−1) we obtain error bounds of the form ‖ f ( k) − s ( k) ‖ ∞ ≃ O( h n + 1 − k ) ( k = 0, …, n).