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

This article is cited in 16 scientific papers (total in 16 papers)

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
Received: 08.06.1970

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
\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
\jour Math. USSR-Izv.
\yr 1971
\vol 5
\issue 1
\pages 97--129

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    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  mathnet  crossref  mathscinet  zmath
    2. V. L. Velikin, “Precise approximation values by Hermitian splines on classes of differentiable function”, Math. USSR-Izv., 7:1 (1973), 163–184  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath
    4. N. P. Korneichuk, “On extremal problems in the theory of best approximation”, Russian Math. Surveys, 29:3 (1974), 7–43  mathnet  crossref  mathscinet  zmath
    5. V. N. Temlyakov, “Asymptotic behavior of best approximations of continuous functions”, Math. USSR-Izv., 11:3 (1977), 551–569  mathnet  crossref  mathscinet  zmath
    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  mathnet  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  mathscinet  zmath
    8. S. M. Nikol'skii, “Aleksandrov and Kolmogorov in Dnepropetrovsk”, Russian Math. Surveys, 38:4 (1983), 41–55  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    9. N. P. Korneichuk, “Duality of extremal problems in function spaces and approximation of functions”, Russian Math. Surveys, 40:4 (1985), 195–196  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    11. S. K. Bagdasarov, “Maximization of functionals in $H^\omega [a,b]$”, Sb. Math., 189:2 (1998), 159–226  mathnet  crossref  crossref  mathscinet  zmath  isi
    12. S. K. Bagdasarov, “Extremal functions of integral functionals in $H^\omega[a,b]$”, Izv. Math., 63:3 (1999), 425–480  mathnet  crossref  crossref  mathscinet  zmath  isi
    13. N. P. Korneichuk, “Best Approximation and Symmetric Decreasing Rearrangements of Functions”, Proc. Steklov Inst. Math., 232 (2001), 172–186  mathnet  mathscinet  zmath
    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  mathnet  crossref  crossref  mathscinet  zmath  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  mathnet  crossref  crossref  mathscinet  isi
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