Best basis selection for approximation in \(L_p\)
This interesting paper on multivariate approximation extends the results of \textit{D.L. Donoho} [Appl. Comput. Harmon. Anal. 1, 100--115 (1993; Zbl 0796.62083)] from \(L_2\) to \(L_p\), \(p\neq 2\). Any extension of approximation results from \(L_2\) to \(L_p\) requires a substitute of the notion of orthonormal bases in \(L_2\). For this purpose, the authors use greedy bases in \(L_p\) (introduced by \textit{S. Konyagin} and \textit{V. Temlyakov} [East J. Approx. 5, 365-379 (1999; Zbl 1084.46509)]) which are democratic and unconditional bases in \(L_p\). In this paper, the authors study the approximation of a function class \({\mathcal F}\) in \(L_p\) by choosing first a basis \(B\) and then using \(n\)-term approximation with the elements of \(B\). Into the competition for best bases, all greedy bases in \(L_p\) are entered. The authors show that if the function class \({\mathcal F}\) is well-oriented with respect to a particular basis \(B\) then, in a certain sense, this basis is the best choice for this type of approximation.
scientific article; zbMATH DE number 1971444