On best \(m\)-term approximations and the entropy of sets in the space \(L^ 1\)
The authors investigate approximate characteristics of sets of the spaces \(L^1 (\mathbb{R}^d)\), \(L^p (\mathbb{R}^d)\), \(p>1\), using results on the geometric properties of finite-dimensional convex bodies. First, for a wide class of systems \(\Phi= \{\varphi_n (x)\}\) they establish lower bounds for best approximations of functions of Sobolev classes of polynomials of the form \[ \sum_{i=1}^m a_{n_i} \varphi_{n_i} (x), \qquad 1\leq n_1< n_2< \dots< n_m; \] here the coefficients and indices depend, in general, on the function approximated. In the second part of the paper they establish lower bounds for the \(\varepsilon\)-entropy, widths, and best \(m\)-term trigonometric approximations in classes of functions of many variables with bounded mixed derivative or difference. Their method gives the possibility of obtaining order-precise lower bound for entropy numbers of the class \(W^r_p\) in the space \(L^q\) for \(p= \infty\), \(q=1\) and even \(r\). In this essential paper the authors prove several interesting results and their proofs are rather complex.
scientific article; zbMATH DE number 830754
scientific article