G. Akishev. On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space ... P. 9-20

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 polyno\-mials. 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

Received March 31, 2019

Revised May 19, 2019

Accepted May 26, 2019

Funding Agency: 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).

Gabdolla Akishev, Dr. Phys.-Math. Sci., Prof., L.N.Gumilyov Eurasian National University, Nur–Sultan, 100008 Republic Kazakhstan; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: akishev_g@mail.ru

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Cite this article as: G.Akishev, On the exactness of the inequality of different metrics for trigonometric polynomials in the generalized Lorentz space, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2019, vol. 25, no. 2, pp. 9–20.