scholarly journals Estimates of the error of interval quadrature formulas on some classes of differentiable functions

2020 ◽  
Vol 28 (1) ◽  
pp. 12
Author(s):  
V.P. Motornyi ◽  
D.A. Ovsyannikov
Keyword(s):  
Quadrature Formula ◽  
Modulus Of Continuity ◽  
Quadrature Formulas ◽  
Periodic Functions ◽  
Differentiable Functions ◽  
The Given

The exact value of error of interval quadrature formulas$$\int_0^{2\pi}f(t)dt -\frac{\pi}{nh}\sum_{k=0}^{n-1}\int_{-h}^hf(t+\frac {2k\pi}{n})dt = R_n(f;\vec{c_0};\vec{x_0};h)$$obtained for the classes $W^rH^{\omega} (r=1,2,...)$ of differentiable periodic functions for which the modulus of continuity of the  $r -$th derivative is majorized by the given modulus of continuity $\omega(t)$. This interval quadrature formula coincides with the rectangles formula for the Steklov functions $f_h(t)$ and is optimal for some important classes of functions.

scholarly journals The best, by coefficients, weighted quadrature formulas of the form $\sum\limits_{k=1}^n \sum\limits_{i=0}^{r-1} A_{ki} f^{(i)}_{jk}$ for some classes of differentiable functions

1987 ◽  
pp. 37
Author(s):  
Ye.Ye. Dunaichuk
Keyword(s):  
Quadrature Formula ◽  
Sharp Estimate ◽  
Quadrature Formulas ◽  
Differentiable Functions ◽  
Derivatives Of ◽  
Weighted Quadrature

For the quadrature formula (with non-negative, integrable on $[0,1]$ function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find the form of the best coefficients $A^0_{ki}$ ($k = \overline{1, n}$, $i = \overline{0, r-1}$) for fixed nodes $\gamma_k$ ($k = \overline{1, n}$) and we give the sharp estimate of the remainder of this formula on the classes $W^r_p$, $r = 1, 2, \ldots$, $1 \leqslant p \leqslant \infty$.

scholarly journals Necessary conditions of optimality of quadrature formulas with weight on some classes of differentiable functions

1987 ◽  
pp. 47
Author(s):  
Ye.Ye. Dunaichuk
Keyword(s):  
Weight Function ◽  
Quadrature Formula ◽  
Necessary Conditions ◽  
Quadrature Formulas ◽  
Necessary Conditions Of Optimality ◽  
Differentiable Functions ◽  
Derivatives Of ◽  
Continuous Weight

For the quadrature formula (with positive, continuous weight function) that is defined by the values of the function and its derivatives of up to and including $(r-1)$-th order, we find necessary conditions of optimality on the classes $W^r_p$, $r = 1, 2, \ldots$, $p = 2$, $p = \infty$.

scholarly journals Simpson type inequalities and applications

2021 ◽  
Author(s):  
Muhammad Uzair Awan ◽  
Muhammad Zakria Javed ◽  
Michael Th. Rassias ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor
Keyword(s):  
Quadrature Formula ◽  
Convex Functions ◽  
Integral Identity ◽  
Higher Order ◽  
Auxiliary Result ◽  
First Order ◽  
New Class ◽  
Differentiable Functions

AbstractA new generalized integral identity involving first order differentiable functions is obtained. Using this identity as an auxiliary result, we then obtain some new refinements of Simpson type inequalities using a new class called as strongly (s, m)-convex functions of higher order of $$\sigma >0$$ σ > 0 . We also discuss some interesting applications of the obtained results in the theory of means. In last we present applications of the obtained results in obtaining Simpson-like quadrature formula.

APPROXIMATIVE PROPERTIES OF ABEL–POISSON-TYPE OPERATORS ON THE GENERALIZED HÖLDER CLASSES

2021 ◽  
Vol 1 ◽  
pp. 76-83
Author(s):  
Yuri I. Kharkevich ◽  
◽  
Alexander G. Khanin ◽  
Keyword(s):  
Modulus Of Continuity ◽  
Applied Mathematics ◽  
Continuous Functions ◽  
Elliptic Type ◽  
Periodic Functions ◽  
French Mathematician ◽  
First Order ◽  
Function Classes ◽  
In The Beginning ◽  
Poisson Type

The paper deals with topical issues of the modern applied mathematics, in particular, an investigation of approximative properties of Abel–Poisson-type operators on the so-called generalized Hölder’s function classes. It is known, that by the generalized Hölder’s function classes we mean the classes of continuous -periodic functions determined by a first-order modulus of continuity. The notion of the modulus of continuity, in turn, was formulated in the papers of famous French mathematician Lebesgue in the beginning of the last century, and since then it belongs to the most important characteristics of smoothness for continuous functions, which can describe all natural processes in mathematical modeling. At the same time, the Abel-Poisson-type operators themselves are the solutions of elliptic-type partial differential equations. That is why the results obtained in this paper are significant for subsequent research in the field of applied mathematics. The theorem proved in this paper characterizes the upper bound of deviation of continuous -periodic functions determined by a first-order modulus of continuity from their Abel–Poisson-type operators. Hence, the classical Kolmogorov–Nikol’skii problem in A.I. Stepanets sense is solved on the approximation of functions from the classes by their Abel–Poisson-type operators. We know, that the Abel–Poisson-type operators, in partial cases, turn to the well-known in applied mathematics Poisson and Jacobi–Weierstrass operators. Therefore, from the obtained theorem follow the asymptotic equalities for the upper bounds of deviation of functions from the Hölder’s classes of order from their Poisson and Jacobi–Weierstrass operators, respectively. The obtained equalities generalize the known in this direction results from the field of applied mathematics.

scholarly journals On multidimensional Jackson's theorem

2021 ◽  
pp. 30
Author(s):  
S.A. Pichugov
Keyword(s):  
Modulus Of Continuity ◽  
Periodic Functions ◽  
Linear Polynomial ◽  
Multiple Variables ◽  
Polynomial Methods ◽  
Methods Of Approximation

We have found the best linear polynomial methods of approximation of continuous periodic functions of multiple variables in uniform metric with concave modulus of continuity.

An optimal quadrature formula in the Sobolev space

2021 ◽  
Vol 65 (3) ◽  
pp. 46-59
Keyword(s):  
Sobolev Space ◽  
Quadrature Formula ◽  
Approximate Calculation ◽  
Order Of Convergence ◽  
Quadrature Formulas ◽  
Explicit Formulas ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Quadrature Formulas

This paper studies the problem of construction of optimal quadrature formulas for approximate calculation of integrals with trigonometric weight in the L(2m)(0, 1) space for any ω ൐= 0, ω ∈ R. Here explicit formulas for the optimal coefficients are obtained. We study the order of convergence of the optimal formulas for the case m = 1, 2. The obtained optimal quadrature formulas are exact for Pm−1(x), where Pm−1(x) is a polynomial of degree (m − 1).

A Quadrature Formula for a Hypersingular Integral Containing the Weight Function of Jacobi Polynomials with Complex Exponents

2020 ◽  
pp. 94-100
Author(s):  
A.V. Sahakyan
Keyword(s):  
Weight Function ◽  
Integral Equations ◽  
Quadrature Formula ◽  
Jacobi Polynomials ◽  
Quadrature Formulas ◽  
Hölder Continuous ◽  
Approximate Methods ◽  
Hypersingular Integral ◽  
Hypersingular Integral Equations ◽  
True Value

Although the concept of a hypersingular integral was introduced by Hadamard at the beginning of the 20th century, it began to be put into practical use only in the second half of the century. The theory of hypersingular integral equations has been widely developed in recent decades and this is due to the fact that they describe the governing equations of many applied problems in various fields: elasticity theory, fracture mechanics, wave diffraction theory, electrodynamics, nuclear physics, geophysics, theory vibrator antennas, aerodynamics, etc. It is analytically possible to calculate the hypersingular integral only for a very narrow class of functions; therefore, approximate methods for calculating such an integral are always in the field of view of researchers and are a rapidly developing area of computational mathematics. There are a very large number of papers devoted to this subject, in which various approaches are proposed both to approximate calculation of the hypersingular integral and to the solution of hypersingular integral equations, mainly taking into account the specifics of the behavior of the densi-ty of the hypersingular integral. In this paper, quadrature formulas are obtained for a hypersingular integral whose density is the product of the Hölder continuous function on the closed interval [–1, 1], and weight function of the Jacobi polynomials . It is assumed that the exponents α and β can be arbitrary complex numbers that satisfy the condition of non-negativity of the real part. The numerical examples show the convergence of the quadrature formula to the true value of the hypersingular integral. The possibility of applying the mechanical quadrature method to the solution of various, including hypersingular, integral equations is indicated.

Best quadrature formula on the class W * r L2 of periodic functions

1974 ◽  
Vol 16 (2) ◽  
pp. 701-708
Author(s):  
N. E. Lushpai
Keyword(s):  
Quadrature Formula ◽  
Periodic Functions
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