Best quadrature formulas calculation of curvilinear integrals for some classes of functions and currves

Abstract

For an approximate calculation of a curvilinear integral
$$J(f;\Gamma):=\int\limits_{\Gamma}f(x_1,x_2,\ldots,x_m)dt$$
when the curve $\Gamma$ is given by parametric equations
$$x_{1}=\varphi_{1}(t),
x_{2}=\varphi_{2}(t),\ldots,x_{m}=\varphi_{m}(t), 0\leq t\leq L$$
the quadrature formula is entered into consideration
$$J(f;\Gamma):\approx\sum_{k=1}^{N}p_{k}\, f\Bigl(\varphi_{1}(t_k),\,
\varphi_{2}(t_k), \ldots,\, \varphi_{m}(t_k)\Bigr),$$ where
$P=\left\{p_{k}\right\}_{k=1}^{N}$ and $T:=\left\{t_{k}:0\leq
t_{1}<t_{2}<\cdots<t_{N}\leq L\right\}$-- are arbitrary vector
coefficients and nodes. Let
$H^{\omega_{1},\ldots,\omega_{m}}[0,L]$-- sets of curves $\Gamma$,
whose coordinate functions $\varphi_{i}(t)\in H^{\omega_{i}}[0,L] \
(i=\overline{1,m}),$ where $\omega_{i}(t) \ (i=\overline{1,m})$--
are given moduli of continuity $\mathfrak{M}_{\rho}^{\omega,p}$--
functions class $f(M),$ defined in point $M\in\Gamma,$ such for any
two points
$M^{\prime}=M(x_{1}^{\prime},x_{2}^{\prime},\ldots,x_{m}^{\prime}),$
$M^{\prime\prime}=M(x_{1}^{\prime\prime},x_{2}^{\prime\prime},\ldots,x_{m}^{\prime\prime})$
belonging to a curve $\Gamma \in
H^{\omega_{1},\ldots,\omega_{m}}[0,L]$ satsify the condition
$$\Bigl|f(M^{\prime})-f(M^{\prime\prime})\Bigr|\le\omega(\rho_{p}(M^{\prime},
M^{\prime\prime})),$$ where $$\rho_{p}(M^{\prime},
M^{\prime\prime})=\left\{\sum_{i=1}^{m}|x^{\prime}_{i}-x_{i}^{\prime\prime}|^{p}\right\}^{1/p},
\ 1\leq p\leq \infty,$$ $\omega(t)$-- given moduls of continuity.
It is proved that among all quadrature formulas of the above from,
the best for a class of functions $\mathfrak{M}_{\rho}^{\omega,p}$
and a class of curves $H^{\omega_{1},\ldots,\omega_{m}}[0,1]$, is
the formula of average rectangles.

The exact error estimate of the best quadrature formula is
calculated for all the functional classes under consideration and
the curves are given a generalization for more general classes of
functions.}