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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)
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UDC: 517.51
MSC: 42A05, 42A10, 46E30
Received: 31.03.2019

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}


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