Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method
The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W 2 ( m ,0) by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here...
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| Published in | Acta mathematica Sinica. English series Vol. 37; no. 7; pp. 1066 - 1088 |
|---|---|
| Main Authors | , , |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.07.2021
Springer Nature B.V |
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| Abstract | The paper studies Sard’s problem on construction of optimal quadrature formulas in the space
W
2
(
m
,0)
by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space
W
2
(
m
,0)
is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue
D
m
(
hβ
) of the differential operator
d
2
m
d
x
2
m
−
1
is constructed. Further, Sobolev’s method of construction of optimal quadrature formulas in the space
W
2
(
m
,0)
, which based on the discrete analogue
D
m
(
hβ
), is described. Next, for
m
= 1 and
m
= 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space
W
2
(3,0)
for the cases
m
= 1 and
m
= 3 are presented. |
|---|---|
| AbstractList | The paper studies Sard’s problem on construction of optimal quadrature formulas in the space
W
2
(
m
,0)
by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space
W
2
(
m
,0)
is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue
D
m
(
hβ
) of the differential operator
d
2
m
d
x
2
m
−
1
is constructed. Further, Sobolev’s method of construction of optimal quadrature formulas in the space
W
2
(
m
,0)
, which based on the discrete analogue
D
m
(
hβ
), is described. Next, for
m
= 1 and
m
= 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space
W
2
(3,0)
for the cases
m
= 1 and
m
= 3 are presented. The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W2(m,0) by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W2(m,0) is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue Dm(hβ) of the differential operator d2mdx2m−1 is constructed. Further, Sobolev’s method of construction of optimal quadrature formulas in the space W2(m,0), which based on the discrete analogue Dm(hβ), is described. Next, for m = 1 and m = 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W2(3,0) for the cases m = 1 and m = 3 are presented. |
| Author | Hayotov, Abdullo Shadimetov, Kholmat Boltaev, Aziz |
| Author_xml | – sequence: 1 givenname: Aziz surname: Boltaev fullname: Boltaev, Aziz organization: V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences – sequence: 2 givenname: Abdullo surname: Hayotov fullname: Hayotov, Abdullo email: hayotov@mail.ru organization: V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences – sequence: 3 givenname: Kholmat surname: Shadimetov fullname: Shadimetov, Kholmat organization: V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent Railway Engineering Institute |
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| Cites_doi | 10.1016/j.cam.2012.11.010 10.1007/s10092-013-0076-6 10.1007/BF02575942 10.1007/s11075-016-0150-7 10.1007/s10986-014-9244-x 10.1137/0703025 10.1007/s11075-010-9440-7 10.3103/S0003701X10010020 10.1016/j.apnum.2012.08.004 10.1002/sapm1950291118 10.1090/S0025-5718-1974-0341825-3 10.1007/978-94-015-8913-0 10.1016/j.amc.2014.07.033 10.11948/2017076 10.1007/s10092-019-0320-9 10.1007/978-3-0348-5836-6 10.1007/978-0-387-34149-1_35 10.17516/1997-1397-2018-11-3-383-396 10.2307/2372095 10.1016/j.amc.2015.12.022 10.1007/s10092-013-0080-x 10.1016/j.cam.2010.07.021 10.1016/j.amc.2015.02.093 10.1090/S0002-9904-1964-11054-5 10.1137/0702012 |
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| Keywords | discrete analogue 65D30 65D32 optimal coefficients the error functional The extremal function 41A15 Sobolev’s method 41A05 |
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S., Sirojiddinov, S. H.: The Theory of Complex Variable Functions (in Uzbek), Fan, Tashkent, 1976 Boltaev, A. K.: An extremal function of an optimal quadrature formula (in Russian). Uzbek. Math. Zh., no. 2, 57–65 (2011) KöhlerPOn the weights of Sard’s quadrature formulasCalcolo198825169186105375310.1007/BF02575942 BabuškaIOptimal quadrature formulas (in Russian)Dokladi Akad. Nauk SSSR1963149227229 SobolevS LVaskevichV LThe Theory of Cubature Formulas1997DordrechtKluwer Academic Publishers Group10.1007/978-94-015-8913-0 ShadimetovK MHayotovA RNuralievF AOn an optimal quadrature formula in Sobolev space L2(m)(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_2^{(m)}(0,1)$$\end{document}J. Comput. Appl. Math.201324391112300337510.1016/j.cam.2012.11.010 Nikol’skiiS MTo question about estimation of approximation by quadrature formulas (in Russian)Uspekhi Matem. Nauk195051651770036.323012(36) Shadimetov, Kh. M.: The discrete analogue of the differential operator d2m/dx2m and its construction. Questions of Computations and Applied Mathematics (in Russian), Tashkent, 22–35, (1985), arXiv: 1001.0556 [math.NA] Shadimetov, Kh. M., Hayotov, A. R.: Properties of the discrete analogue of the differential operator d2m/dx2m − d2m−2/dx2m−2 (in Russian). Uzbek Math. Zh., no. (4), 7 A Sard (9506_CR28) 1949; 71 9506_CR36 P Köhler (9506_CR21) 1988; 25 9506_CR33 9506_CR34 K M Shadimetov (9506_CR43) 2013; 243 A R Hayotov (9506_CR19) 2015; 259 A R Hayotov (9506_CR15) 2014; 54 I Babuška (9506_CR4) 1963; 149 N D Boltaev (9506_CR9) 2017; 7 A Cabada (9506_CR10) 2014; 244 K M Shadimetov (9506_CR41) 2013; 53 9506_CR37 9506_CR38 I J Schoenberg (9506_CR31) 1966; 3 9506_CR46 D V Ionescu (9506_CR20) 1957 A R Hayotov (9506_CR16) 2018; 11 I J Schoenberg (9506_CR30) 1965; 2 I J Schoenberg (9506_CR32) 1974; 28 A A Maljukov (9506_CR23) 1976 G Coman (9506_CR12) 1972; 17 9506_CR49 V S Vladimirov (9506_CR48) 1979 N R Avezova (9506_CR2) 2010; 46 A R Hayotov (9506_CR18) 2014; 51 K M Shadimetov (9506_CR44) 2016; 276 N D Boltaev (9506_CR8) 2017; 74 9506_CR50 S M Nikol’skii (9506_CR27) 1988 S L Sobolev (9506_CR45) 1974 K M Shadimetov (9506_CR42) 2014; 51 S L Sobolev (9506_CR47) 1997 9506_CR51 9506_CR6 J H Ahlberg (9506_CR1) 1967 L F Meyers (9506_CR25) 1950; 29 A K Boltaev (9506_CR7) 2014; 2 K M Shadimetov (9506_CR35) 2002; 5 9506_CR3 S M Nikol’skii (9506_CR26) 1950; 5 I J Schoenberg (9506_CR29) 1964; 70 F Lanzara (9506_CR22) 2000; 5 A Ghizzetti (9506_CR14) 1970 9506_CR24 A R Hayotov (9506_CR17) 2011; 57 K M Shadimetov (9506_CR39) 2011; 235 T Catinaş (9506_CR11) 2006; 51 G Coman (9506_CR13) 1972; 6 P Blaga (9506_CR5) 2007; 52 K M Shadimetov (9506_CR40) 2012; 62 |
| References_xml | – reference: Babaev, S. S., Hayotov, A. R.: Optimal interpolation formulas in the space W2(m,m−1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W_2^{(m,m - 1)}$$\end{document}. Calcolo, 56(3), Paper No. 23, 25pp. (2019) – reference: Shadimetov, Kh. 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| Snippet | The paper studies Sard’s problem on construction of optimal quadrature formulas in the space
W
2
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by Sobolev’s method. This problem consists of two... The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W2(m,0) by Sobolev’s method. This problem consists of two parts:... |
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| SubjectTerms | Differential equations Lagrange multiplier Linear equations Mathematical analysis Mathematics Mathematics and Statistics Operators (mathematics) Quadratures Trigonometric functions |
| Title | Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev’s Method |
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