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 Optimal quadrature formulae and minimal monosplines in Lq

1970 ◽  
Vol 11 (1) ◽  
pp. 48-56 ◽  
Author(s):  
J. Kautsky
Keyword(s):  
Quadrature Formula ◽  
Quadrature Formulae ◽  
Optimal Quadrature ◽  
Derivatives Of ◽  
Optimal Quadrature Formulae ◽  
Optimal Formula

SummaryThe quadrature formula of order m using values of derivatives up to the m — 1st order with the best possible bound in is derived. Using certain properties of the polynomials minimal in Lq norm, it is proved that the optimal formula not use the derivatives of m — 1st order if m is even.

scholarly journals Calculation of Chakalov-Popoviciu quadratures of Radau and Lobatto type

2002 ◽  
Vol 43 (3) ◽  
pp. 429-447 ◽  
Author(s):  
Miodrag M. Spalević
Keyword(s):  
Numerical Method ◽  
Quadrature Formula ◽  
Spline Function ◽  
Finite Interval ◽  
Analytic Formula ◽  
Numerical Results ◽  
Quadrature Formulae

AbstractA numerical method for calculation of the generalized Chakalov-Popoviciu quadrature formulae of Radau and Lobatto type, using the results given for the generalized Chakalov-Popoviciu quadrature formula, is given. Numerical results are included. As an application we discuss the problem of approximating a function f on the finite interval I = [a, b] by a spline function of degree m and variable defects dv, with n (variable) knots, matching as many of the initial moments of f as possible. An analytic formula for the coefficients in the generalized Chakalov-Popoviciu quadrature formula is given.

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

Order-Preserving Approximations to Successive Derivatives of Periodic Functions by Iterated Splines

1988 ◽  
Vol 25 (6) ◽  
pp. 1442-1452 ◽  
Author(s):  
M. J. Shelley ◽  
G. R. Baker
Keyword(s):  
Periodic Functions ◽  
Derivatives Of

scholarly journals Inequalities for norms of derivatives of non-periodic functions with non-symmetric constraints on higher derivatives

2012 ◽  
Vol 20 ◽  
pp. 99
Author(s):  
V.A. Kofanov
Keyword(s):  
Real Line ◽  
Periodic Functions ◽  
Higher Derivatives ◽  
Derivatives Of

For non-periodic functions $x \in L^r_{\infty}(\mathbb{R})$ defined on the whole real line we established the analogs of certain inequality of V.F. Babenko.

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 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 $W^r_p$, $r = 1, 2, \ldots$, $1 \leqslant p \leqslant \infty$.

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