scholarly journals On the best interval quadrature formula for the class of functions $W^3 H^{\omega}$

2021 ◽  
Vol 19 ◽  
pp. 47
Author(s):  
Ye.V. Derets
Keyword(s):  
Quadrature Formula ◽  
Modulus Of Continuity ◽  
Optimal Interval ◽  
Third Derivative ◽  
Class Of Functions

The optimal interval quadrature formula in the class of functions with convex majorant of the modulus of continuity of third derivative is obtained.

scholarly journals On error of interval quadrature formula of rectangles for the class $W^2 H_1^{\omega}$

2021 ◽  
Vol 15 ◽  
pp. 84
Author(s):  
Ye.V. Derets
Keyword(s):  
Quadrature Formula ◽  
Modulus Of Continuity ◽  
Second Derivative ◽  
Integral Modulus ◽  
Class Of Functions

We found the error of interval quadrature formula of rectangles for the class of functions with given convex majorant of integral modulus of continuity of second derivative.

scholarly journals On the best interval quadrature formula for the class $W^2 H_1^{\omega}$

2021 ◽  
Vol 16 ◽  
pp. 73
Author(s):  
Ye.V. Derets
Keyword(s):  
Quadrature Formula ◽  
Modulus Of Continuity ◽  
Second Derivative ◽  
Integral Modulus ◽  
Class Of Functions

We prove that the rectangular interval quadrature formula is the best for the class of functions with given convex majorant of integral modulus of continuity of the second derivative.

scholarly journals Convergence of a quadrature formula for variable-signed weight functions

1998 ◽  
Vol 57 (2) ◽  
pp. 275-288
Author(s):  
H.S. Jung ◽  
K.H. Kwon
Keyword(s):  
Weight Function ◽  
Rate Of Convergence ◽  
Quadrature Formula ◽  
Modulus Of Continuity ◽  
Higher Order ◽  
Weight Functions ◽  
Quadratic Mean ◽  
Mean Convergence ◽  
Interpolating Polynomials

A quadrature formula for a variable-signed weight function w(x) is constructed using Hermite interpolating polynomials. We show its mean and quadratic mean convergence. We also discuss the rate of convergence in terms of the modulus of continuity for higher order derivatives with respect to the sup norm.

scholarly journals On optimal interval quadrature formulae on classes of differentiable periodic functions

2021 ◽  
Vol 15 ◽  
pp. 16
Author(s):  
V.F. Babenko ◽  
D.S. Skorokhodov
Keyword(s):  
Quadrature Formula ◽  
Invariant Set ◽  
Periodic Functions ◽  
Quadrature Formulae ◽  
Optimal Interval ◽  
Arbitrary Permutation ◽  
Derivatives Of

We solved the problem about the best interval quadrature formula on the class $W^r F$ of differentiable periodic functions with arbitrary permutation-invariant set $F$ of derivatives of order $r$. We proved that the formula with equal coefficients and $n$ node intervals, which have equidistant middle points, is the best on given class.

scholarly journals About Nodal Systems for Lagrange Interpolation on the Circle

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
E. Berriochoa ◽  
A. Cachafeiro ◽  
J. M. García Amor
Keyword(s):  
Unit Circle ◽  
Modulus Of Continuity ◽  
Lagrange Interpolation ◽  
Continuous Functions ◽  
Trigonometric Interpolation ◽  
Laurent Polynomials ◽  
Class Of Functions

We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by thenroots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on[-1,1]and the Lagrange trigonometric interpolation are obtained.

scholarly journals Dunkl analogue of Szász-mirakjan operators of blending type

2018 ◽  
Vol 16 (1) ◽  
pp. 1344-1356 ◽  
Author(s):  
Sheetal Deshwal ◽  
P.N. Agrawal ◽  
Serkan Araci
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Integral Form ◽  
Degree Of Approximation ◽  
Modulus Of Smoothness ◽  
Approximation Properties ◽  
Korovkin Theorem ◽  
Dunkl Analogue ◽  
Derivatives Of ◽  
Class Of Functions

AbstractIn the present work, we construct a Dunkl generalization of the modified Szász-Mirakjan operators of integral form defined by Pǎltanea [1]. We study the approximation properties of these operators including weighted Korovkin theorem, the rate of convergence in terms of the modulus of continuity, second order modulus of continuity via Steklov-mean, the degree of approximation for Lipschitz class of functions and the weighted space. Furthermore, we obtain the rate of convergence of the considered operators with the aid of the unified Ditzian-Totik modulus of smoothness and for functions having derivatives of bounded variation.

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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