Approximation of space curves by polygonal lines in $L_p$
We consider the class $H^{\omega_{1},\omega_{2},\ldots,\omega_{m}}$ of parametric curves in the $m$-dimensional Euclidean space whose coordinate curves belong to the classes $H^{\omega_{i}}[0,L]$ $(i=\overline{1,m})$, respectively; i.e., their moduli of continuity are dominated by the functions $\omega_{i}$. We solve the problem of finding an upper bound for the mutual deviation in the norm of the space $L_{p}[0,L]$ $(1\le p<\infty)$ of two curves from this class under the condition that they intersect at $N$ $(N\ge2)$ points of the interval $[0,L]$. We also find the exact value for the upper bound of the deviation in the $L_{p}$ metric of a curve $\Gamma$ belonging to a class $H^{\omega_{1},...,\omega_{m}}$ defined by upper convex moduli of continuity $\omega_{i}(t)$, $i=\overline{1,m}$, from an interpolation polygonal line inscribed in this curve with $N$ $(N\ge2)$ interpolation nodes. The obtained results generalize V.F. Storchai's result on the approximation of continuous functions by interpolation polygonal lines in the metric of the space $L_p[0,L]$ $(1\le p\le\infty)$.