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 Izv. Akad. Nauk SSSR Ser. Mat., 1971, Volume 35, Issue 1, Pages 93–124 (Mi izv1921)

Extreme values of functionals and best approximation on classes of periodic functions

N. P. Korneichuk

Abstract: In this paper, we compute upper bounds for the best approximation by trigonometric polynomials in the metrics $C$ and $L$ on the classes $W^rH_\omega$ of $2\pi$-periodic functions such that $|f^{(r)}(x')-f^{(r)}(x")|\leqslant\omega(|x'-x"|)$, where $\omega(t)$ is a given convex modulus of continuity. In doing this, we obtain a series of results which explain certain new properties of differentiable functions expressed in terms of rearrangements. Also, we obtain precise estimates for functionals of the form $\int_0^{2\pi}fg dx$, where $f\in H_\omega$, and $g$ belongs to a certain class of differentiable functions defined by bounds on the norm of $g$ and its derivatives in $C$ or $L$.

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English version:
Mathematics of the USSR-Izvestiya, 1971, 5:1, 97–129

Bibliographic databases:

UDC: 517.5
MSC: 42A04, 42A08

Citation: N. P. Korneichuk, “Extreme values of functionals and best approximation on classes of periodic functions”, Izv. Akad. Nauk SSSR Ser. Mat., 35:1 (1971), 93–124; Math. USSR-Izv., 5:1 (1971), 97–129

Citation in format AMSBIB
\Bibitem{Kor71} \by N.~P.~Korneichuk \paper Extreme values of functionals and best approximation on classes of periodic functions \jour Izv. Akad. Nauk SSSR Ser. Mat. \yr 1971 \vol 35 \issue 1 \pages 93--124 \mathnet{http://mi.mathnet.ru/izv1921} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=277988} \zmath{https://zbmath.org/?q=an:0216.39101} \transl \jour Math. USSR-Izv. \yr 1971 \vol 5 \issue 1 \pages 97--129 \crossref{https://doi.org/10.1070/IM1971v005n01ABEH001015} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. N. P. Korneichuk, “Inequalities for differentiable periodic functions and best approximation of one class of functions by another”, Math. USSR-Izv., 6:2 (1972), 417–428
2. V. L. Velikin, “Precise approximation values by Hermitian splines on classes of differentiable function”, Math. USSR-Izv., 7:1 (1973), 163–184
3. V. P. Motornyi, “On the best quadrature formula of the form $\sum_{k=1}^np_kf(x_k)$ for some classes of differentiable periodic functions”, Math. USSR-Izv., 8:3 (1974), 591–620
4. N. P. Korneichuk, “On extremal problems in the theory of best approximation”, Russian Math. Surveys, 29:3 (1974), 7–43
5. V. N. Temlyakov, “Asymptotic behavior of best approximations of continuous functions”, Math. USSR-Izv., 11:3 (1977), 551–569
6. A. I. Stepanets, “Estimates of the deviations of partial Fourier sums on classes of continuous periodic functions of several variables”, Math. USSR-Izv., 17:2 (1981), 369–403
7. N. P. Korneichuk, “Widths in $L_p$ of classes of continuous and of differentiable functions, and optimal methods of coding and recovering functions and their derivatives”, Math. USSR-Izv., 18:2 (1982), 227–247
8. S. M. Nikol'skii, “Aleksandrov and Kolmogorov in Dnepropetrovsk”, Russian Math. Surveys, 38:4 (1983), 41–55
9. N. P. Korneichuk, “Duality of extremal problems in function spaces and approximation of functions”, Russian Math. Surveys, 40:4 (1985), 195–196
10. N. P. Korneichuk, “S. M. Nikol'skii and the development of research on approximation theory in the USSR”, Russian Math. Surveys, 40:5 (1985), 83–156
11. S. K. Bagdasarov, “Maximization of functionals in $H^\omega [a,b]$”, Sb. Math., 189:2 (1998), 159–226
12. S. K. Bagdasarov, “Extremal functions of integral functionals in $H^\omega[a,b]$”, Izv. Math., 63:3 (1999), 425–480
13. N. P. Korneichuk, “Best Approximation and Symmetric Decreasing Rearrangements of Functions”, Proc. Steklov Inst. Math., 232 (2001), 172–186
14. V. F. Babenko, N. V. Parfinovich, “Exact Values of Best Approximations for Classes of Periodic Functions by Splines of Deficiency 2”, Math. Notes, 85:4 (2009), 515–527
15. S. K. Bagdasarov, “Kolmogorov inequalities for functions in classes $W^rH^\omega$ with bounded $\mathbb L_p$-norm”, Izv. Math., 74:2 (2010), 219–279
16. V. F. Babenko, N. V. Parfinovich, “On the Exact Values of the Best Approximations of Classes of Differentiable Periodic Functions by Splines”, Math. Notes, 87:5 (2010), 623–635
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