A.A. Shabozova. Approximation of space curves by polygonal lines in $L_p$ ... P. 311-318

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 $L_p[0,L]$ $(1\le p\le\infty)$.

Keywords: parametric curves, modulus of continuity, interpolation broken lines.

The paper was received by the Editorial Office on May 10, 2017

Adolat Azamovna Shabozova, doctoral student, Tajik National University, Dushanbe, 734025 Tajikistan,
e-mail: shabozova91@mail.ru .


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