Estimates of M-term Approximations of Functions of Several Variables in the Lorentz Space by a Constructive Method

Abstract

In the paper, the Lorentz space \(L_{q,\tau}(\mathbb{T}^m)\) of periodic functions of several variables, the Nikol'skii–Besov class \(S_{q,\tau,\theta}^{r} B\) and the associated class \(W_{q,\tau,\theta}^{a,b,r}\) for \(1 < q, \tau < \infty, 1 \leq \theta < \infty\) are considered. Estimates are established for the best \(M\)-term trigonometric approximations of functions of the classes \(W_{q,\tau,\theta}^{a,b,r}\) and \(S_{q,\tau,\theta}^{r} B\) in the norm of the space \(L_{p,\tau}(\mathbb{T}^m)\) for different relations between the parameters \(q, \tau, p, \tau_2, a, \theta\). The proofs of the theorems are based on the constructive method developed by V.N. Temlyakov.