Abstract
We obtain characterizations (and prove the corresponding equivalence of norms) of function spaces B ( $$ \mathbb{I} $$ k ) and L ( $$ \mathbb{I} $$ 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 B ( $$ \mathbb{I} $$ ) and L ( $$ \mathbb{I} $$ 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, ..., s n ) ∈ ℝ + , 1 ≤ p, q, r ≤ ∞, m = (m 1, ..., m n ) ∈ ℕ n , k = m 1 +... + m n , and $$ \mathbb{I} $$ = ℝ or $$ \mathbb{T} $$ ). In the periodic case, we study the Fourier widths of these function classes.