Direct and inverse approximation theorems for local trigonometric bases

We investigate the approximation of smooth functions by local trigonometric bases. In particular, we are interested in the local behavior of the approximation error in the L^p-norm. We derive direct and inverse approximation theorems that describe the best approximation on an interval by a finite linear combination of basis functions with support in this interval. As a result, we characterize the Besov spaces on an interval as approximation spaces with respect to a local trigonometric basis. These local results are generalized to the approximation on the real line by linear combinations which are locally finite. The proofs are based on the classical inequalities of Jackson and Timan which are applied to local trigonometric bases by the means of folding and unfolding operators.