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 Optimal quadrature formula in the sense of Sard in K2(P3) space

2014 ◽  
Vol 95 (109) ◽  
pp. 29-47 ◽  
Author(s):  
Abdullo Hayotov ◽  
Gradimir Milovanovic ◽  
Kholmat Shadimetov
Keyword(s):  
Hilbert Space ◽  
Sobolev Space ◽  
Quadrature Formula ◽  
Asymptotic Optimality ◽  
Trigonometric Functions ◽  
Numerical Examples ◽  
Optimal Quadrature Formula ◽  
Optimal Quadrature ◽  
Optimal Coefficients ◽  
Optimal Formula

We construct an optimal quadrature formula in the sense of Sard in the Hilbert space K2(P3). Using Sobolev?s method we obtain new optimal quadrature formula of such type and give explicit expressions for the corresponding optimal coefficients. Furthermore, we investigate order of the convergence of the optimal formula and prove an asymptotic optimality of such a formula in the Sobolev space L (3)2 (0, 1). The obtained optimal quadrature formula is exact for the trigonometric functions sin x, cos x and for constants. Also, we include a few numerical examples in order to illustrate the application of the obtained optimal quadrature formula.

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.

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