RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy Inst. Mat. i Mekh. UrO RAN: Year: Volume: Issue: Page: Find

 Trudy Inst. Mat. i Mekh. UrO RAN, 2019, Volume 25, Number 2, Pages 9–20 (Mi timm1619)

On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space

G. A. Akishevab

a L. N. Gumilev Eurasian National University, Astana
b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

Abstract: We consider the generalized Lorentz space $L_{\psi,\tau}(\mathbb{T}^m)$ defined by some continuous concave function $\psi$ such that $\psi (0)=0$. For two spaces $L_{\psi_1,\tau_1}(\mathbb{T}^m)$ and $L_{\psi_2,\tau_2}(\mathbb{T}^{m})$ such that $\alpha_{\psi_{1}}={\underline\lim}_{t\rightarrow 0}\psi_{1}(2t)/\psi_{1}(t) = \beta_{\psi_{2}} = \overline{\lim}_{t\rightarrow 0}\psi_{2}(2t)/\psi_{2}(t)$, we prove an order-exact inequality of different metrics for multiple trigonometric polynomials. We also prove an auxiliary statement for functions of one variable with monotonically decreasing Fourier coefficients in a trigonometric system. In this statement we establish a two-sided estimate for the norm of the function $f\in L_{\psi, \tau}(\mathbb{T})$ in terms of the series composed of the Fourier coefficients of this function.

Keywords: generalized Lorentz space, Jackson–Nikol'skii inequality, trigonometric polynomial.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 02.A03.21.0006 This work was supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

DOI: https://doi.org/10.21538/0134-4889-2019-25-2-9-20

Full text: PDF file (227 kB)
First page: PDF file
References: PDF file   HTML file

UDC: 517.51
MSC: 42A05, 42A10, 46E30

Citation: G. A. Akishev, “On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space”, Trudy Inst. Mat. i Mekh. UrO RAN, 25, no. 2, 2019, 9–20

Citation in format AMSBIB
\Bibitem{Aki19} \by G.~A.~Akishev \paper On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2019 \vol 25 \issue 2 \pages 9--20 \mathnet{http://mi.mathnet.ru/timm1619} \crossref{https://doi.org/10.21538/0134-4889-2019-25-2-9-20} \elib{http://elibrary.ru/item.asp?id=38071594}