Geodesic
Mixed series in ultraspherical polynomials and
Sbornik. Mathematics, Tome 194 (2003) no. 3, pp. 423-456 
I. I. Sharapudinov. Mixed series in ultraspherical polynomials and. Sbornik. Mathematics, Tome 194 (2003) no. 3, pp. 423-456. http://geodesic.mathdoc.fr/item/SM_2003_194_3_a5/
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     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_3_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

New (mixed) series in ultraspherical polynomials $P_n^{\alpha,\alpha}(x)$ are introduced. The basic difference between a mixed series in the polynomials $P_n^{\alpha,\alpha}(x)$ and a Fourier series in the same polynomials is as follows: a mixed series contains terms of the form $\dfrac{2^rf_{r,k}^\alpha}{(k+2\alpha)^{[r]}}P_{k+r}^{\alpha-r,\alpha-r}(x)$, where $1\leqslant r$ is an integer and $f_{r,k}^\alpha$ is the $k$ th Fourier coefficient of the derivative $f^{(r)}(x)$ with respect to the ultraspherical polynomials $P_k^{\alpha,\alpha}(x)$. It is shown that the partial sums ${\mathscr Y}_{n+2r}^\alpha(f,x)$ of a mixed series in the polynomial $P_k^{\alpha,\alpha}(x)$ contrast favourably with Fourier sums $S_n^\alpha(f,x)$ in the same polynomials as regards their approximation properties in classes of differentiable and analytic functions, and also in classes of functions of variable smoothness. In particular, the ${\mathscr Y}_{n+2r}^\alpha(f,x)$ can be used for the simultaneous approximation of a function $f(x)$ and its derivatives of orders up to $(r- 1)$, whereas the $S_n^\alpha(f,x)$ are not suitable for this purpose.

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

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

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

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

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

[6] Sharapudinov I. I., “Priblizhenie funktsii s peremennoi gladkostyu summami Fure–Lezhandra”, Matem. sb., 191:5 (2000), 143–160 | MR | Zbl

[7] Segë G., Ortogonalnye mnogochleny, Fizmatgiz, M., 1962

[8] Gasper G., “Positivity and special functions”, Theory and application of special functions, Proc. Adv. Semin. (Madison, 1975), ed. R. A. Askey, Academic Press, New York, 1975, 375–433 | MR

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

[10] Akhiezer N. I., Lektsii po teorii approksimatsii, Nauka, M., 1965 | MR