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

Abstract

We 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.