We establish exact-order estimates for the norms of generalized derivatives of the Dirichlet-type kernels with an arbitrary choice of harmonics in the space L q , 2 < q < 1.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
G. H. Hardy and J. E. Littlewood, “A new proof of a theorem of rearrangements,” J. London Math. Soc., 23, No. 91, 163–168 (1948).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
A. I. Stepanets, Methods of Approximation Theory [in Russian], Vols. 1, 2, Institute of Mathematics, Ukrainian National Academy of Sciences, Kiev (2002).
V. N. Temlyakov, Approximation of Periodic Functions, Nova Science Publishers, New York (1993).
S. V. Konyagin, “On the Littlewood problem,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 2, 243–265 (1981).
O. C. McGehe, L. Pigno, and B. Smith, “Hardy inequality and L1-norm of exponential sums,” Ann. Math., 113, No. 3, 613–618 (1981).
V. M. Tikhomirov, “Approximation theory,” in: VINITI Series on Contemporary Problems of Mathematics (Fundamental Trends) [in Russian], Vol. 14, VINITI, Moscow (1987), pp. 103–260.
V. E. Maiorov, “Bernstein–Nikols’kii inequalities and estimates for the norms of the Dirichlet kernels for trigonometric polynomials with arbitrary harmonics,” Mat. Zametki, 47, No. 6, 55–61 (1990).
É. S. Belinskii, “Approximation by trigonometric polynomials of given length on classes of functions with bounded mixed derivative and some extremal problems,” in: Proc. of the Fourth Saratov Winter School “Theory of Functions and Approximations” (January 25–February 5, 1988) [in Russian], Part 2, Saratov University, Saratov (1990), pp. 43–45.
É. S. Belinskii, “Two extremal problems for trigonometric polynomials with given number of harmonics,” Mat. Zametki, 49, No. 1, 12–18 (1991).
E. M. Galeev, “Approximation of periodic functions of one and several variables,” in: Constructive Theory of Functions (Varna, 1987), Publ. House Bulgar. Acad. Sci., Sofia (1988), pp. 138–144.
É. S. Belinskii and E. M. Galeev, “On the least value of norms of the mixed derivatives of trigonometric polynomials with given number of harmonics,” Vestn. Mosk. Gos. Univ., Ser. 1. Mat. Mekh., 2, 3–7 (1991).
E. M. Galeev, “Order estimates for the derivatives of periodic multidimensional Dirichlet 𝛼-kernel in a mixed norm,” Mat. Sb., 117(159), No. 1, 32–43 (1982).
H. M. Vlasyk, “Estimates for the norms of generalized derivatives of Dirichlet-type kernels with an arbitrary choice of harmonics,” in: Differential Equations and Related Problems of Analysis [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv, 13, No. 2 (2016), pp. 88–100.
A. S. Romanyuk, “Inequalities for the L p -norms of (ψ, β)-derivatives and Kolmogorov widths of the classes of functions of many variables \( {L}_{\beta, p}^{\psi }, \)” in: Investigations in the Approximation Theory of Functions [in Russian], Proc. of the Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1987), pp. 92–105.
S. M. Nikol’skii, Approximation of Functions of Many Variables and Embedding Theorems [in Russian], Nauka, Moscow (1989).
D. Jackson, “Certain problems of closest approximation,” Bull. Amer. Math. Soc., 39, No. 12, 889–906 (1933).
A. Zygmund, Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, No. 10, pp. 1310–1323, October,2017.
Rights and permissions
About this article
Cite this article
Vlasyk, H.M. Order Estimates of the L q -Norms of Generalized Derivatives of the Dirichlet-Type Kernels with an Arbitrary Choice of Harmonics. Ukr Math J 69, 1520–1536 (2018). https://doi.org/10.1007/s11253-018-1453-8
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-018-1453-8