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 .

REFERENCES

1.    Sendov Bl. Haussdorff Approximation. NY etc., Springer Publ., 1990, 388 p.

2.   Zavyalov Yu.S, Kvasov B.I., Miroshnichencko V.L. Metodi splain funksiy [Methods of Spline-Functions], Moscow,  Nauka Publ., 1980, 352 p.

3.   Martynyuk V.T. Approximation by polygonal lines of curves given by parametric equations in the Hausdorff metric. Ukr. Math. J., 1976, vol. 28, no. 1, pp. 68-72.

4.   Nazarenko N.A. Local recovery curves by parametric splines. Geometric theory of functions and topology, Kiev, 1981, pp. 55-62 (in Russian).

5.   Vakarchuk S.B. Approximation by spline-curves of curves given in parametric form. Ukr. Math. J., 1983, vol. 35, no. 3, pp. 303-306.

6.   Vakarchuk S.B. Exact constants for the approximation of plane curves by polynomial curves and polygonal lines. Izvestiya VUZ. Mathematika, 1988, vol. 32, no. 2, pp. 19-26.

7.   Korneichuk N.P. Optimal coding of vector-functions. Ukr. Math. J., 1988, vol. 40, no. 6, pp. 621-627.

8.   Korneichuk N.P. Approximation and optimal coding of smooth plane curves. Ukr. Math. J., 1989, vol. 41, no. 4, pp. 429-435.

9.   Shabozov M.Sh., Shabozova A.A. Approximating curves by broken lines. Vestnik St. Petersburg University, ser. 1, iss. 2, pp. 68-76 (in Russian).

10.   Nikolskiy S.M. Kvadraturnie formuli [Quadrature Formulae]. Moscow, Nauka Publ., 1986, 256 p.

11.   Storchai V.F. The diviation of polygonal functions in the $L_{p}$ metric. Math. Notes, 1969, vol. 5, no. 1, pp. 21-25.

12.   Shabozova A.A. Polygonal interpolation of curves in the space $\mathbb{R}^{m}$. Izvestiya TSU, 2015, iss. 4, pp.107-112 (in Russian).