On bivariate smoothness spaces associated with nonlinear approximation (Q706345)

It is known that classical linear smoothness spaces fail to characterize multivariate nonlinear piecewise polynomial approximation. In this paper certain nonlinear smoothness spaces are proposed that come close to providing such a characterization. Let \(\Omega \subset {\mathbb R}^2\) be a bounded domain. Consider a partition \(\Lambda \) of \(\Omega\) formed by \(n_E=n_E(\Lambda)\) non-intersecting curves \(b_j: [0,1] \to \Omega\), each of finite length len\((b_j)\). The curves partition \(\Omega\) into \(n_F\) open connected subdomains \(\Omega_k\). With each \(b_j\) a parameter \(t_j \in (0,1]\) is associated, and \(\Lambda (t)\),\(t>0\), denotes the set of all \(\Lambda\) for which \(\sum_{j=1}^{n_E} t_j^{-1} \leq t^{-1}\). For \(f \in L_p(\Omega)\), \(1 \leq p<\infty\) and \(r_1, r_2 \in {\mathbb N}\), let \[ \tilde{K}_{r_1,r_2}(f,t)_p:=\inf_{\Lambda \in \Lambda (t)} \left (\sum_1^{n_E} \text{ len}(b_j) K_{r_1}(b_j, t_j^{r_1})_{\infty ,1} + \sum_{k=1}^{n_F} K_{r_2}(f,\Omega_k)_p^p \right )^{1/p}, \] where \(K_{r_1}\) and \(K_{r_2}\) are certain \(K\)-functionals associated with the curves \(b_j\) and the subdomains \(\Omega_k\). A function \(f \in L_p({\mathbb R}^2)\), \(1 \leq p<\infty\), is said to be in the \(\tilde{B}\)-space \({\tilde{B}}_q^{\alpha,r_1,r_2} (L_p(\Omega))\), \(0<q<\infty\), if \[ (f)_{\tilde{B}}:=\left (\int_0^1(t^{-\alpha}\tilde{K}_{r_1,r_2}(f,t)_p)^q {dt \over t} \right )^{1/q}<\infty, \] with the standard extension to \(q=\infty\). The authors study the relations between the classical Besov spaces and the \(\tilde {B}\) spaces. They establish Jackson-type inequalities for piecewise polynomial approximation, with a particular attention to approximation over triangles. They also study relations between the \(\tilde{K}\) and the so-called Mumford-Shah functionals.