Orders of approximation by local exponential splines

Yu. S. Volkov   ; E. G. Pytkeev   ; V. T. Shevaldin

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 18 (2012) no. 4, pp. 135-144

Résumé

We continue the study of approximation properties of local exponential splines on a uniform grid with step $h>0$ corresponding to a linear differential operator $\mathcal L$ with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E. V. Strelkova and V. T. Shevaldin). We find order estimates as $h\to0$ for the error of approximation of certain Sobolev classes of functions by the mentioned splines, which are exact on the kernel of the operator $\mathcal L$.

Détail

Keywords: approximation, local exponential splines, order estimates.

Bibliographie

[1] Zavyalov Yu. S., Kvasov B. I., Miroshnichenko V. L., Metody splain-funktsii, Nauka, M., 1980, 352 pp. | MR

[2] Lyche T., Schumaker L. L., “Local spline approximation methods”, J. Approx. Theory, 15:4 (1975), 294–325 | DOI | MR | Zbl

[3] Shevaldina E. V., “Approksimatsiya lokalnymi $\mathcal L$-splainami chetnogo poryadka, sokhranyayuschimi yadro differentsialnogo operatora”, Izv. TulGU. Estestvennye nauki, 2 (2009), 62–73

[4] Shevaldina E. V., “Lokalnye $\mathcal L$-splainy, sokhranyayuschie yadro differentsialnogo operatora”, Sib. zhurn. vychisl. matematiki, 13:1 (2010), 111–121 | Zbl

[5] Strelkova E. V., Shevaldin V. T., “Approksimatsiya lokalnymi $\mathcal L$-splainami, tochnymi na podprostranstvakh yadra differentsialnogo operatora”, Tr. In-ta matematiki i mekhaniki UrO RAN, 16, no. 4, Ekaterinburg, 2010, 272–280

[6] Strelkova E. V., Shevaldin V. T., “Formosokhranenie pri approksimatsii lokalnymi eksponentsialnymi splainami proizvolnogo poryadka”, Tr. In-ta matematiki i mekhaniki UrO RAN, 17, no. 3, Ekaterinburg, 2011, 291–299

[7] Shevaldin V. T., “Approksimatsiya lokalnymi parabolicheskimi splainami s proizvolnym raspolozheniem uzlov”, Sib. zhurn. vychisl. matematiki, 8:1 (2005), 77–88 | Zbl

[8] Subbotin Yu. N., “Approksimatsiya polinomialnymi i trigonometricheskimi splainami tretego poryadka, sokhranyayuschaya nekotorye svoistva approksimiruemykh funktsii”, Tr. In-ta matematiki i mekhaniki UrO RAN, 13, no. 2, Ekaterinburg, 2007, 156–166

[9] Volkov Yu. S., Strelkova E. V., Shevaldin V. T., “Lokalnaya approksimatsiya splainami so smescheniem uzlov”, Mat. trudy, 14:2 (2011), 73–82 | MR

[10] Shevaldin V. T., “Nekotorye zadachi ekstremalnoi interpolyatsii v srednem dlya lineinykh differentsialnykh operatorov”, Tr. MIAN SSSR, 164, 1983, 203–240 | MR | Zbl

[11] Sharma A., Tsimbalario I., “Nekotorye lineinye differentsialnye operatory i obobschennye raznosti”, Mat. zametki, 21:2 (1977), 161–172 | MR | Zbl

[12] ter Morsche H. G., Interpolation and extremal properties of $\mathcal L$-spline functions, Dissertation, Techinische Hogeschool Eindhoven, Eindhoven, 1982, 124 pp. | MR | Zbl

[13] Korneichuk N. P., Splainy v teorii priblizheniya, Nauka, M., 1984, 352 pp. | MR

[14] Prasolov V. V., Mnogochleny, MTsIMO, M., 2003, 336 pp.