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

AbstractWe obtain characterizations (and prove the corresponding equivalence of norms) of function spaces Bpqsm ($$ \mathbb{I} $$k) and Lpqsm ($$ \mathbb{I} $$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 $$ \mathcal{W}_m^\mathbb{I} $$ 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 Bpqsm ($$ \mathbb{I} $$) and Lpqsm ($$ \mathbb{I} $$k) by special partial sums of these series in the metric of Lr($$ \mathbb{I} $$k) for a number of relations between the parameters s, p, q, r, and m (s = (s1, ..., sn) ∈ ℝ+n, 1 ≤ p, q, r ≤ ∞, m = (m1, ..., mn) ∈ ℕn, k = m1 +... + mn, and $$ \mathbb{I} $$ = ℝ or $$ \mathbb{T} $$). In the periodic case, we study the Fourier widths of these function classes.