Entropy, universal coding, approximation, and bases properties
Let \({\mathcal E}=\{e_n\}_{n=0}^\infty\) be a Schauder basis of a Banach space \(X\). For \(f=\sum\theta_ie_i \in X\) the linear approximation error is defined by \[ \rho_n(f,{\mathcal E},X)=\left \| \sum_{i=n}^\infty \theta_ie_i \right \| _X , \] and the nonlinear, \(n\)-term, approximation error by \[ \sigma_n(f,{\mathcal E},X)=\inf_h \{\| f-h\| : h=\sum_{i \in \Lambda} \theta_ie_i, \; | \Lambda | \leq n \}. \] For \(0<r,s <\infty\), the space \(B_r^s=B_r^s({\mathcal E},X)\) is defined as the set of all \(f \in X\) for which \[ \left \{ \sum_{n=1}^\infty [n^s\rho_{n-1}(f,{\mathcal E},X)]^r1/n \right \}^{1/r}:=\| f\| _{B_r^s({\mathcal E},X)}<\infty , \] with the usual modification for \(r=\infty\). The space \(A_r^s=A_r^s({\mathcal E},X)\) is defined similarly but with \(\sigma_n(f,{\mathcal E},X)\) instead of \(\rho_n(f,{\mathcal E},X)\). The authors establish some metric entropy results assuming that the normalized basis \({\mathcal E}\) is quasi-greedy and verifies Temlyakov's \(p\)-property for some \(p>0\). They prove that the \(\varepsilon\)-entropy in \(X\) of the unit ball \(U_r^s\) of the space \(B_r^s\) is \(\asymp \varepsilon ^{-1/s}\) while for \(0<\delta<s\) the \(\varepsilon\)-entropy of the intersection of \(U_r^\delta\) with the unit ball \(V_r^s\) of the spase \(A_r^s\) is \(\asymp \varepsilon ^{-1/s}\log(1/\varepsilon).\) Thus the metric entropy distinguishes between linear and nonlinear approximation. The lower bounds for \(\varepsilon\)-entropy are obtained using estimates for large deviations in binomial and hypergeometric ditributions. The upper bounds are based on some special universal coding. Extending and refining Donoho's results for Hilbert spaces, the authors prove that under certain conditions spaces \(A_\infty^r\) are characterized by the metric entropy behavior. The last section of the paper concerns applications to wavelets in \(L_p\).
scientific article; zbMATH DE number 2101898