On best harmonic synthesis of periodic functions. (English. Russian original) Zbl 1336.65208
J. Math. Sci., New York 209, No. 1, 115-129 (2015); translation from Fundam. Prikl. Mat. 18, No. 5, 155-174 (2013).
Summary: In this paper, we construct optimal methods of recovery of periodic functions from a known (exact or inexact) finite family of their Fourier coefficients. The proposed approach to constructing recovery methods is compared with the approach based on the Tikhonov regularization method.
MSC:
| 65T40 | Numerical methods for trigonometric approximation and interpolation |
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
References:
| [1] | V. A. Il’in and E. G. Poznyak, Foundations of Mathematical Analysis, Part II (in Russian), Nauka, Moscow (1973). |
| [2] | G. G. Magaril-Il’yaev and K. Yu. Osipenko, “Optimal recovery of functions and their derivatives from Fourier coefficients prescribed with an error,” Sb. Math., 193, No. 3, 387-407 (2002). · Zbl 1060.42003 · doi:10.1070/SM2002v193n03ABEH000637 |
| [3] | G. G. Magaril-Il’yaev and K. Yu. Osipenko, “Optimal recovery of functions and their derivatives from inaccurate information about the spectrum and inequalities for derivatives,” Funct. Anal. Appl., 37, No. 3, 203-214 (2003). · Zbl 1048.41007 · doi:10.1023/A:1026084617039 |
| [4] | G. G. Magaril-Il’yaev and K. Yu. Osipenko, “Optimal recovery of values of functions and their derivatives from inaccurate data on the Fourier transform,” Sb. Math., 195, No. 10, 1461-1476 (2004). · Zbl 1072.42002 · doi:10.1070/SM2004v195n10ABEH000853 |
| [5] | G. G. Magaril-Il’yaev and K. Yu. Osipenko, “On optimal harmonic synthesis from inaccurate spectral data,” Funct. Anal. Appl., 44, No. 3, 223-225 (2010). · Zbl 1271.42002 · doi:10.1007/s10688-010-0029-7 |
| [6] | G. G. Magaril-Il’yaev and K. Yu. Osipenko, “How best to recover a function from its inaccurately given spectrum?” Math. Notes, 92, No. 1, 51-58 (2012). · Zbl 1268.90125 · doi:10.1134/S0001434612070061 |
| [7] | G. G. Magaril-Il’yaev and E. O. Sivkova, “Best recovery of the Laplace operator of a function from incomplete spectral data,” Sb. Math., 203, No. 4, 569-580 (2012). · Zbl 1251.49045 · doi:10.1070/SM2012v203n04ABEH004235 |
| [8] | E. O. Sivkova, “On optimal recovery of the Laplacian of a function from its inaccurately given Fourier transform,” Vladikavkaz. Mat. Zh., 14, No. 4, 63-72 (2012). · Zbl 1326.42018 |
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