Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I

We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces $\mathbf B^{sm}_{pq}(\mathbb I^k)$ and $\mathbf L^{sm}_{pq}(\mathbb I^k)$ of Nikol'skii–Besov and Lizorkin–Triebel types, respectively, in terms of representations of functions in these spaces by Fourier series with respect to a multiple system $\mathcal W^\mathbb I_m$ of Meyer wavelets and in terms of sequences of the Fourier coefficients with respect to this system. We establish order-sharp estimates for the approximation of functions in $B^{sm}_{pq}(\mathbb I^k)$ and $L^{sm}_{pq}(\mathbb I^k)$ by special partial sums of these series in the metric of $L_r(\mathbb I^k)$ for a number of relations between the parameters $s,p,q,r$, and $m$ ($s=(s_1,\dots,s_n)\in\mathbb R^n_+$, $1\leq p,q,r\leq\infty$, $m=(m_1,\dots,m_n)\in\mathbb N^n$, $k=m_1+\dots+m_n$, and $\mathbb I= \mathbb R$ or $\mathbb T$). In the periodic case, we study the Fourier widths of these function classes.