STUDIA MATHEMATICA 159 (2) (2003) Convergence of greedy approximation II. The trigonometric system by S. V. Konyagin (Moscow) and V. N. Temlyakov (Columbia, SC) Abstract. We study the following nonlinear method of approximation by trigono- metric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form G m (f ) := ∑ k∈Λ  f (k)e i(k,x) , where Λ ⊂ Z d is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients  f (k) of the function f . Note that G m (f ) gives the best m-term approximant in the L 2 -norm, and therefore, for each f ∈ L 2 , ‖f − G m (f )‖ 2 → 0 as m →∞. It is known from previous results that in the case of p = 2 the condition f ∈ L p does not guarantee the convergence ‖f − G m (f )‖ p → 0 as m →∞. We study the following question. What conditions (in addition to f ∈ L p ) provide the convergence ‖f − G m (f )‖ p → 0 as m →∞? In the case 2 <p ≤∞ we find necessary and sufficient conditions on a decreasing sequence {A n } ∞ n=1 to guarantee the L p -convergence of {G m (f )} for all f ∈ L p satisfying a n (f ) ≤ A n , where {a n (f )} is the decreasing rearrangement of the absolute values of the Fourier coefficients of f . 1. Introduction. We study the following natural nonlinear method of summation of trigonometric Fourier series. Consider a periodic function f ∈ L p (T d ), 1 ≤ p ≤∞ (L ∞ (T d )= C (T d )), defined on the d-dimensional torus T d . Let m ∈ N and t ∈ (0, 1] be given, and let Λ m be a set of k ∈ Z d with the properties: (1.1) min k∈Λ m |  f (k)|≥ t max k∈Λ m |  f (k)|, |Λ m | = m, where  f (k) := (2π) −d T d f (x)e −i(k,x) dx is the kth Fourier coefficient of f . We define G t m (f ) := G t m (f, T ) := S Λ m (f ) :=  k∈Λ m  f (k)e i(k,x) 2000 Mathematics Subject Classification : 42B08, 42B35. This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003. [161]