Approximation of Curves by Broken Lines in Lp

In the present work, the exact value of the upper bound of the deviation is found in the Lp metric for curves Γ determined by parametric equations in an m-dimensional Euclidean space depending on inscribed interpolation broken lines for the class $${{H}^{{{{\omega }_{1}}, \ldots ,{{\omega }_{m}}}}}$$ defined either by arbitrary moduli of continuity ωi(t), i = $$\overline {1,m} $$ , or by convex moduli. The problem of obtaining the upper bounds of deviation of parametrically-defined curves Γ, G ∈ $${{H}^{{{{\omega }_{1}},{{\omega }_{2}} \ldots ,{{\omega }_{m}}}}}$$ , the coordination functions φi(t) and ψi(t) (i = $$\overline {1,m} $$ ) of which (respectively) belong to the class $${{H}^{{{{\omega }_{i}}}}}$$ [0, L] (i = $$\overline {1,m} $$ ) and intersect at N (N ≥ 2) points of the partition of segment [0, L], is solved. The obtained results represent an extension of the findings of V. F. Storchay on the approximation of continuous functions by interpolation broken lines in the metric of space Lp[0, L] (1 ≤ p ≤ ∞).