Abstract:
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.
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