Geodesic
Approximation of functions of variable smoothness by Fourier–Legendre sums
Sbornik. Mathematics, Tome 191 (2000) no. 5, pp. 759-777 
I. I. Sharapudinov. Approximation of functions of variable smoothness by Fourier–Legendre sums. Sbornik. Mathematics, Tome 191 (2000) no. 5, pp. 759-777. http://geodesic.mathdoc.fr/item/SM_2000_191_5_a6/
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     author = {I. I. Sharapudinov},
     title = {Approximation of functions of variable smoothness by {Fourier{\textendash}Legendre} sums},
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     volume = {191},
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     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_5_a6/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Assume that $0<\mu\leqslant 1$, and let $r\geqslant 1$ be an integer. Let $\Delta =\{a_1,\dots,a_l\}$, where the $a_i$ are points in the interval $(-1,1)$. The classes $S^rH^\mu_\Delta$ and $S^rH^\mu_\Delta(B)$ are introduced. These consist of functions with absolutely continuous $(r-1)$th derivative on $[-1,1]$ such that their $r$th and $(r+1)$th derivatives satisfy certain conditions outside the set $\Delta$. It is proved that for $0<\mu<1$ the Fourier–Legendre sums realize the best approximation in the classes $S^rH^\mu_\Delta(B)$. Using the Fourier–Legendre expansions, polynomials $\mathscr Y_{n+2r}$ of order $n+2r$ are constructed that possess the following property: for $0<\mu<1$ the $\nu$th derivative of the polynomial $\mathscr Y_{n+2r}$ approximates $f^{(\nu)}(x)$ $(f\in S^rH^\mu_\Delta)$ on $[-1,1]$ to within $O(n^{\nu+1-r-\mu})$, and the accuracy is of order $O(n^{\nu-r-\mu})$ outside $\Delta$.

[1] Gronwall T., “Über die Laplacesche Reiche”, Math. Ann., 74 (1913), 213–270 | DOI | MR | Zbl

[2] Rau H., “Über die Lebesgueschen Konstanten der Reihenentwicklugen nach Jacobischen Polynomen”, J. für Math., 161 (1929), 237–254 | Zbl

[3] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[4] Agakhanov S. A., Natanson G. I., “Funktsiya Lebega summ Fure–Yakobi”, Vestnik LGU. Ser. matem., mekh., astron., 1:1 (1968), 11–23 | MR | Zbl

[5] Agakhanov S. A., Natanson G. I., “Priblizhenie funktsii summami Fure–Yakobi”, Dokl. AN SSSR, 166:1 (1966), 9–10 | MR | Zbl

[6] Badkov V. M., “Otsenki funktsii Lebega i ostatka ryada Fure–Yakobi”, Sib. matem. zhurn., 9:6 (1968), 1263–1283 | MR | Zbl

[7] Badkov V. M., “Approksimativnye svoistva ryadov Fure po ortogonalnym mnogochlenam”, UMN, 33:4 (202) (1978), 51–106 | MR | Zbl

[8] Sharapudinov I. I., “O nailuchshem priblizhenii i summakh Fure–Yakobi”, Matem. zametki, 34:5 (1983), 651–661 | MR | Zbl

[9] Gasper G., “Positivity and special functions”, Theory Appl. Spec. Funct., Proc. Adv. Semin. (Madison), 1975, 375–433 | MR | Zbl

[10] Timan A. F., Teoriya priblizheniya funktsii deistvitelnogo peremennogo, Fizmatgiz, M., 1960