The article considers the Lorentz space $L_{q, \tau}(\mathbb{T}^{m})$ of periodic functions of $m$ variables and the class $W_{q, \tau}^{a, b(\cdot), \overline{r}}$ for $1<q, \tau <\infty$, $a>0$, $b(t)$ is a slowly varying function on $[1, \, \infty )$. $W_{q, \tau}^{a, b(\cdot), \overline{r}}$ the class of all functions $f\in L_{q, \tau}(\mathbb{T}^{m})$ for which $S_{n}^{(\overline\gamma)}(f,\overline{x})$ the partial sum over the step hyperbolic cross of the Fourier series in the norm of $L_{q, \tau}(\mathbb{T}^{m})$ converges at rate $2^{-na}b(2^{n})$ as $n\rightarrow \infty$. The main result is the exact order of the best $n$-term trigonometric approximations of functions from the class $W_{q, \tau_{1}}^{a, b(\cdot), \overline{r}}$ in the norm of the space $L_{p, \tau_{2}}(\mathbb{T}^{m})$ in the case $1<q<p\leqslant 2$, for some relations between the parameters $a$, $\tau_{1}$, $\tau_{2}$. The result is proved by a constructive method.
Keywords: Lorentz space, trigonometric system, best $n$-term approximation, constructive method
Received April 30, 2025
Revised September 9, 2025
Accepted September 21, 2025
Funding Agency: The work was carried out with the financial support of the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (grant no. AP19677486).
Gabdolla Akishev, Dr. Phys.-Math. Sci., Prof., Lomonosov Moscow University, Kazakhstan Branch, Astana, 100001 Kazakhstan; Institute of Mathematics and Mathematical Modeling, Almaty, 050010 Kazakhstan, e-mail: akishev_g@mail.ru
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Cite this article as: G. Akishev. On estimates of $n$-term approximations of functions in Lorentz space. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2025, vol. 31, no. 4, pp. 10–25.
