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Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I. (English. Russian original) Zbl 1219.42025
Proc. Steklov Inst. Math. 269, 2-24 (2010); translation from Trudy Mat. Inst. Steklova 269, 8-30 (2010).
Characterizations and special norm equivalences of function spaces $$B_{pq}^{sm} (I^{k})$$ and $$L_{pq}^{sm}(I^{k})$$ of Nikol’skii-Besov and Lizorkin-Triebel types are studied, where $$I$$ denotes the real numbers $${\mathbb R}$$ or the torus. More exactly, wavelet expansions using Meyer wavelets are considered, and the equivalence of norms in these function spaces with discrete norms of sequences of wavelet coefficients is shown.
The author establishes order-sharp estimates for the approximation of functions in $$B_{pq}^{sm} (I^{k})$$ and $$L_{pq}^{sm}(I^{k})$$ by special partial sums of these series in the metric $$L_{r}(I^{k})$$ for a number of relations between the parameters $$s,p,q,r$$, and $$m$$, where $$s=(s_{1}, \dots , s_{n}) \in {\mathbb R}_{+}^{n}$$, $$1 \leq p,q,r \leq \infty$$, $$m= (m_{1}, \dots , m_{n}) \in {\mathbb N}^{n}$$, and $$k=m_{1}+ \dots + m_{n}$$. In the periodic case, the Fourier widths of these function classes are studied.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42C15 General harmonic expansions, frames
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