Asymptotically best quadrature formulas on classes of differentiable functions

1983 ◽  
Vol 35 (1) ◽  
pp. 84-87
Author(s):  
A. A. Ligun ◽  
V. F. Storchai
Keyword(s):  
Quadrature Formulas ◽  
Differentiable 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 On some Hermite–Hadamard inequalities involving k-fractional operators

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shanhe Wu ◽  
Sajid Iqbal ◽  
Muhammad Aamir ◽  
Muhammad Samraiz ◽  
Awais Younus
Keyword(s):  
Fractional Integrals ◽  
Quadrature Formulas ◽  
Fractional Operators ◽  
Differentiable Functions ◽  
Error Estimations

AbstractThe main objective of this paper is to establish some new Hermite–Hadamard type inequalities involving k-Riemann–Liouville fractional integrals. Using the convexity of differentiable functions some related inequalities have been proved, which have deep connection with some known results. At the end, some applications of the obtained results in error estimations of quadrature formulas are also considered.

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 A Note on Hermite–Hadamard–Fejer Type Inequalities for Functions Whose n-th Derivatives Are m-Convex or (α,m)-Convex Functions

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 16
Author(s):  
Sanja Kovač
Keyword(s):  
Convex Function ◽  
Convex Functions ◽  
Remainder Term ◽  
Quadrature Formulas ◽  
Differentiable Functions ◽  
Special Case

In this paper, we develop some Hermite–Hadamard–Fejér type inequalities for n-times differentiable functions whose absolute values of n-th derivatives are (α,m)-convex function. The results obtained in this paper are extensions and generalizations of the existing ones. As a special case, the generalization of the remainder term of the midpoint and trapezoidal quadrature formulas are obtained.

scholarly journals Double condensation of singularities for product-quadrature formulas with differentiable functions

2012 ◽  
Vol 28 (1) ◽  
pp. 83-91
Author(s):  
ALEXANDRU I. MITREA ◽  
Keyword(s):  
Quadrature Formulas ◽  
Triangular Matrices ◽  
Differentiable Functions ◽  
Equidistant Nodes ◽  
The Matrix

The main goal of this paper is to emphasize the phenomenon of double condensation of singularities for spaces of differentiable functions with respect to product-quadrature formulas associated to a class of node triangular matrices in [−1, 1], including a Gegenbauer node matrix and the matrix of equidistant nodes in [−1, 1].

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.

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