Greedy Algorithm and m -Term Trigonometric Approximation

Abstract.

We study the following nonlinear method of approximation by trigonometric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form $G_m(f) := \sum_{k \in \Lambda} \hat f(k) e^{i(k,x)}$ , where $\Lambda \subset {\bf Z}^d$ is a set of cardinality m containing the indices of the m biggest (in absolute value) Fourier coefficients $\hat f(k)$ of function f . We compare the efficiency of this method with the best m -term trigonometric approximation both for individual functions and for some function classes. It turns out that the operator G _m provides the optimal (in the sense of order) error of m -term trigonometric approximation in the L _p -norm for many classes.