Nonlinear approximation of function spaces of mixed smoothness
Let \(\mathbb{N}\) be the set of all positive integers, and let \(X\) be a real Banach space with norm \(\|\cdot\|_X\). If \(\Phi = \{\phi_i\}_{i \in \mathbb{N}}\) is a basis for \(X\) then for each \(f \in X\) there exists a unique sequence of numbers \(\{c_k(f)\}_{k \in \mathbb{N}}\) such that \[ f = \sum_{k=1}^{\infty}c_k(f) \phi_k. \] Let \(C_k(f, X, \Phi) = \|c_k(f) \phi_k\|_X, k \in \mathbb{N}\) and \(\Lambda_m\) be the set of \(m\) natural numbers such that \[ \min_{k \in \Lambda_m} C_k(f, X, \Phi) \geq \max\limits_{j \in \mathbb{N} \setminus \Lambda_m} C_j(f, X, \Phi). \] The \(X\)-greedy algorithm with respect to \(\Phi\) is defined by \[ G^{X}_{m}(f, \Phi) = \sum_{k \in \Lambda_m}c_k(f) \phi_k. \] For a given subspace \(F \subset X\) define \[ G^{X}_{m}(F, \Phi) = \sup\limits _{f \in F}\|f- G^{X}_{m}(f, \Phi)\|_X. \] In the paper under review, the author studied the approximative properties of greedy algorithms with respect to \(X = L^q(\mathbb{T}^d)\) in the Nikoĺskiĭ-Besov classes \(SB_{p \;\theta}^{r}p \theta(\mathbb{T}^d)\) and the Lizorkin-Triebel classes \(SF_{p \;\theta}^{r}p \theta(\mathbb{T}^d)\) of functions of mixed smoothness with respect to an orthonormal system associated with multiple periodic wavelets, constituted by shifts of multiple Dirichlet kernels.
scientific article; zbMATH DE number 6526917
scientific article