GREEDY ALGORITHM AND M-TERM TRIGONOMETRIC APPROXIMATION

We study the following nonlinear method of approximation by trigonomet ric polynomials in this paper. For a periodic function f we take as an approximant a trigonometric polynomial of the form G(m) (f) := Sigma( k epsilon Lambda) (f) over cap(k)e(i(k,x)), where Lambda subset of Z(d ) is a set of cardinality m containing the indices of the nl biggest ( in absolute value) Fourier coefficients (f) over cap(k) of function f. We compare the efficiency of this method with the best m-term trigono metric approximation both for individual functions and for some functi on 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.