Approximation of curves by broken lines in Lp

Abstract

In this paper was found the exact values of upper bounds deviation in L p[0, L ] (1 p < ∞) metrics of curve Γ, defined by parametric equations in n -dimensional space of inscribed in its at the points t k = kL/N, k = 0, N a broken line on the H ω1,...,ωm class given both as an arbitrary or convex modulus of continuity ω i( t ), i = 1, m. The problem of finding the upper bounds of deviation of parametric givencurves Γ, G ∈ Hω1,ω2,...,ωm coordinate functions ϕi(t) and ψi(t) (i = 1, m) of which respectively belong to the class Hωi [0, L] (i = 1, m) intersect in N (N ≥ 2) points of the partition to the segment [0, L].The obtained results are generalizations of the result of V. F. Storchai on the approximation of continuous functions by interpolation polygonal lines in the metric of the space Lp[0, L] (1 ≤ p ≤ ∞). Refs 16.