On the simultaneous approximation of functions and their derivatives by Chebyshev polynomials orthogonal on uniform grid

I. I. Sharapudinov ; T. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 4 (2015), pp. 74-117.

Résumé

The article is dedicated to investigation of approximative properties of polynomial operator $\mathcal{ X}_{m,N}(f)=\mathcal{ X}_{m,N}(f,x)$, which is defined in the space $C[-1,1]$ and based on the use of only discrete values of the function $f(x)$, given in the nodes of uniform grid $\{x_j=-1+jh\}_{j=0}^{N+2r-1}\subset [-1,1]$. This operator can be used for solving the problem of simultaneous approximation of a differentiable function $f(x)$ and its multiple derivatives $f'(x), \ldots, f^{(p)}(x)$. Construction of operators $\mathcal{ X}_{m,N}(f)$ is based on Chebyshev polynomials $T_n^{\alpha,\beta}(x,N)$ $(0\le n\le N-1)$, which form an orthogonal system on the set $\Omega_N=\{0,1,\ldots,N-1\}$ with weight $$ \mu(x)=\mu(x;\alpha,\beta,N)=c{\Gamma(x+\beta+1) \Gamma(N-x+\alpha)\over \Gamma(x+1)\Gamma(N-x)}, $$ i.e. $$ \sum_{x\in\Omega_N}\mu(x)T_n^{\alpha,\beta}(x,N)T_m^{\alpha,\beta}(x,N) =h_{n,N}^{\alpha,\beta}\delta_{nm}. $$ There were obtained upper bounds for the Lebesgue functions of an operator $\mathcal{ X}_{m,N}(f)=\mathcal{ X}_{m,N}(f,x)$ and weighted approximations of the following form $$ {|\frac1{h^{\nu}}\Delta_h^\nu\left[ f(x_{j-\nu})-\mathcal{ X}_{n+2r,N}(f,x_{j-\nu})\right]|\over\left(\sqrt{1-x_{j}^2}+{1\over m}\right)^{r-\nu-\frac12}}. $$

Détail

Keywords: Chebyshev polynomials orthogonal on the grid; Chebyshev polynomials of the first kind; approximation of functions and derivatives.

Bibliographie

[1] Telyakovskii S.A., “Dve teoremy o priblizhenii funktsii algebraicheskimi mnogochlenami”, Matematicheskii sbornik, 70:2 (1966), 252–265

[2] Gopengauz I.Z., “K teoreme A.F.Timana o priblizhenii funktsii mnogochlenami na konechnom otrezke”, Matematicheskie zametki, 1:2 (1967), 163–172 | MR | Zbl

[3] Malozemov V.N., Sovmestnoe priblizhenie funktsii i ee proizvodnykh, Izd-vo LGU, L., 1973 | MR

[4] Dzyadyk V.K., Vvedenie v teoriyu ravnomernogo priblizheniya funktsii, Nauka, M., 1977 | MR

[5] Telyakovskii S.A., “Otsenka odnovremennogo priblizheniya funktsii i ikh proizvodnykh summami Fure”, Matematicheskie zametki, 90:3 (2011), 478–480 | DOI | MR | Zbl

[6] Boks Dzh., Dzhenkins G., Analiz vremennykh ryadov: prognoz i upravlenie, v. 1, 2, Mir, M., 1974 | MR

[7] Deich A.M., Metody identifikatsii dinamicheskikh ob'ektov, Energiya, M., 1979

[8] Solodovnikov V.V., Dmitriev A.N., Egupov N.D., Spektralnye metody rascheta i proektirovaniya sistem upravleniya, Mashinostroenie, M., 1986 | MR

[9] Demidovich B.P., Maron I.A., Osnovy vychislitelnoi matematiki, Nauka, Moskva, 1966 | MR

[10] Stechkin C.B., Subbotin Yu.N., Splainy v vychislitelnoi matematike, Nauka, Moskva, 1976 | MR

[11] Vitushkin A.G., Otsenka slozhnosti zadachi tabulirovaniya, Fizmatlit, Moskva, 1959 | MR

[12] Chebyshev P.L., “O nepreryvnykh drobyakh (1855)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, Moskva, 1947, 103–126

[13] Chebyshev P.L., “Ob odnom novom ryade”, Poln. sobr. soch., v. 2, Izd. AN SSSR, Moskva, 1947, 236–238 | MR

[14] Chebyshev P.L., “Ob interpolirovanii po sposobu naimenshikh kvadratov (1859)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, Moskva, 1947, 314–334 | MR

[15] Chebyshev P.L., “Ob interpolirovanii (1864)”, Poln. sobr. soch., v. 2, Izd. AN SSSR, Moskva, 1947, 357–374 | MR

[16] Chebyshev P.L., “Ob interpolirovanii velichin ravnootstoyaschikh (1875)”, Poln. sobr. soch. Moskva, v. 2, Izd. AN SSSR, 1947, 66–87 | MR

[17] Sharapudinov I.I., “O skhodimosti metoda naimenshikh kvadratov”, Matematicheskie zametki, 53:3 (1993), 131–143 | MR | Zbl

[18] Trefethen L.N., Spectral methods in Matlab, SIAM, Fhiladelphia, 2000 | MR | Zbl

[19] Trefethen L.N., Finite difference and spectral methods for ordinary and partial differential equation, Cornell University, 1996

[20] Arushanyan O.B., Volchenskova N.I., Zaletkin S.F., “O vychislenii koeffitsientov ryadov Chebysheva dlya reshenii obyknovennykh differentsialnykh uravnenii”, Sibirskie elektronnye matematicheskie izvestiya, 8 (2011), 273–283 | MR | Zbl

[21] Saeed Radhoush, Mahmoud Samavat, Mohammad Ali Vali, “Optimal control of linear time-varying systems using the Chebyshev wavelets (a comparative approach)”, Systems Science and Control Engineering: An Open Access Journal, 2 (2014), 691–698 | DOI

[22] Sharapudinov I.I., “Asimptoticheskie svoistva i vesovye otsenki mnogochlenov Chebysheva–Khana algebraicheskimi mnogochlenami”, Matematicheskii sbornik, 183:3 (1991), 408–420

[23] Sharapudinov I.I., “Ob asimptotike mnogochlenov Chebysheva, ortogonalnykh na konechnoi sisteme tochek”, Vestnik MGU. Seriya 1, 1 (1992), 29–35 | MR

[24] Sharapudinov I.I., Mnogochleny, ortogonalnye na diskretnykh setkakh, Izdatelstvo Dag. gos. ped. un-ta, Makhachkala, 1997

[25] Sharapudinov I.I., “Priblizhenie diskretnykh funktsii i mnogochleny Chebysheva, ortogonalnye na ravnomernoi setke”, Matematicheskie zametki, 67:3 (2000), 460–470 | DOI | MR | Zbl

[26] Sharapudinov I.I., “Approksimativnye svoistva operatorov ${\cal Y}_{n+2r}(f)$ i ikh diskretnykh analogov”, Matematicheskie zametki, 72:5 (2002), 765–795 | DOI | MR | Zbl

[27] Sharapudinov I.I., “Smeshannye ryady po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Matematicheskie zametki, 78:3 (2005), 442–465 | DOI | MR | Zbl

[28] Sharapudinov I.I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Lezhandra na klassakh $W^r$”, Matematicheskii sbornik, 197:3 (2006), 135–154 | DOI | MR | Zbl

[29] Sharapudinov T.I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Vestnik Dagestanskogo nauchnogo tsentra RAN, 29 (2007), 12–23