Geodesic
Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels
Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 101-117 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The author considers classes of periodic functions given by variation diminishing convolutions. These classes include, for example, the Sobolev class $W_p^r(\mathbf T)$ and the class $K_p^U(\mathbf T)=\{f|\|U(d/dx)f(\cdot)\|_p\leqslant1\}$, where $U$ is any polynomial with real coefficients and real roots. The Poisson kernel, the de la Vallée–Poussin kernel and many others have the variation diminishing property. For classes given by variation diminishing convolutions the author proves optimality of the rectangle formula among all quadrature formulas of the form $\sum^k_{i=1}\sum^{\nu_i-1}_{j=0}a_{ij}f^{(j)}(x_i)$ with $\sum_{i=1}^k\nu_i\leqslant N$. In addition, a solution of Favard's problem is given and an optimal method of reconstructing functions of these classes is found. Bibliography: 22 titles. 
@article{SM_1987_58_1_a5,
     author = {Nguy\^en Thị Thi\^eu Hoa},
     title = {Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels},
     journal = {Sbornik. Mathematics},
     pages = {101--117},
     year = {1987},
     volume = {58},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1987_58_1_a5/}
}
TY  - JOUR
AU  - Nguyên Thị Thiêu Hoa
TI  - Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels
JO  - Sbornik. Mathematics
PY  - 1987
SP  - 101
EP  - 117
VL  - 58
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1987_58_1_a5/
LA  - en
ID  - SM_1987_58_1_a5
ER  - 
%0 Journal Article
%A Nguyên Thị Thiêu Hoa
%T Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels
%J Sbornik. Mathematics
%D 1987
%P 101-117
%V 58
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1987_58_1_a5/
%G en
%F SM_1987_58_1_a5
Nguyên Thị Thiêu Hoa. Best quadrature formulas and methods of reconstructing functions defined by variation diminishing kernels. Sbornik. Mathematics, Tome 58 (1987) no. 1, pp. 101-117. http://geodesic.mathdoc.fr/item/SM_1987_58_1_a5/

[1] Karlin S., Total positivity, Stanford University Press, Stanford, California, 1968 | MR

[2] Tikhomirov V. M., Nekotorye voprosy teorii priblizhenii, MGU, M., 1976 | MR

[3] Nikolskii S. M., Kvadraturnye formuly, Nauka, M., 1974 | MR

[4] Zigmund A., Trigonometricheskie ryady, Mir, M., 1965 | MR

[5] Krein M. G., Nudelman A. A., Problema momentom Markova i ekstremalnye zadachi, Nauka, M., 1973 | MR

[6] Motornyi V. P., “O nailuchshei kvadraturnoi formule vida $\sum_{k=1}^np_kf(x_k)$ dlya nekotorykh klassov periodicheskikh funktsii”, DAN SSSR, 211:5 (1973), 1060–1062 | MR | Zbl

[7] Zhensykbaev A. A., “Nailuchshaya kvadraturnaya formula dlya nekotorykh klassov periodicheskikh differentsiruemykh funktsii”, Izv. AN SSSR. Ser. matem., 41 (1977), 1110–1124 | Zbl

[8] Zhensykbaev A. A., “Monosplainy minimalnoi normy i nailuchshie kvadraturnye formuly”, UMN, 36:4 (1981), 107–159 | MR | Zbl

[9] Boyanov B. D., “Kharakteristika i suschestvovanie optimalnykh kvadraturnykh formul dlya odnogo klassa differentsiruemykh funktsii”, DAN SSSR, 232:6 (1977), 1233–1236 | MR | Zbl

[10] Oskolkov K. I., “Ob optimalnosti kvadraturnoi formuly s ravnootstoyaschimi uzlami na klassakh periodicheskikh funktsii”, DAN SSSR, 249:1 (1979), 49–52 | MR

[11] Schoenberg I. J., “On variation diminishing integral operators of the convolution type”, Proc. Nat. Acad. Sci., U.S.A., 34 (1948), 164–169 | DOI | MR | Zbl

[12] Mairhuber J. C., Schoenberg I. J., Williamson R. E., “On variation diminishing transformations of the Circle”, Rend. Circ. Mat. Palermo (2), 8 (1959), 241–270 | DOI | MR | Zbl

[13] Pinkus A., “On $n$-width of periodic functions”, J. Analyse Math., 35 (1979), 209–235 | DOI | MR | Zbl

[14] Chakhkiev M. A., “Ob optimalnosti ravnootstoyaschikh uzlov”, DAN SSSR, 264:4 (1982), 836–839 | MR

[15] Babenko V. F., Grankina T. A., “O nailuchshikh kvadraturnykh formulakh na klassakh svertok s $\mathcal{O}(M,\Delta)$-yadrami”, Issledovanie po sovremennym problemam summirovaniya i priblizheniya funktsii i ikh prilozheniyam. Sbornik nauchnykh trudov, DGU, Dnepropetrovsk, 1982, 6–13

[16] Nguen Tkhi Tkheu Khoa, “O nailuchshikh metodakh integrirovaniya i vosstanovleniya funktsii na klassakh, zadavaemykh svertkami, ne uvelichivayuschimi ostsillyatsiyu”, UMN, 39:2 (1984), 177–178 | MR

[17] Favard J., “Sur l'interpolation”, J. Math. Pures Appl., 19 (1940), 281–306 | MR | Zbl

[18] Karlin S., “Some variational problems on certain Sobolev spaces and perfect spline”, Bull. Amer. Math. Soc., 79 (1973), 124–128 | DOI | MR | Zbl

[19] De Boor C., “A remark concerning perfect splines”, Bull. Amer. Math. Soc., 80:4 (1974), 724–727 | DOI | MR | Zbl

[20] Boyanov B. D., Optimalno v'zstnovyavanie na funktsii i funktsionali, Dis. ... dokt. matem. nauk, Sofiiskii universitet, Sofiya, 1980

[21] Smolyak S. A., Ob optimalnom vosstanovlenii funktsii i funktsionalov ot nikh, Dis. ... kand. fiz.-matem nauk, M., 1965

[22] Nguen Tkhi T. Kh., Nailuchshie kvadraturnye formuly i metody vosstanovleniya na klassakh funktsii, zadavaemykh svertkami, ne uvelichivayuschimi ostsillyatsiyu, Dis. ... kand. fiz.-matem. nauk, MGU, M., 1985