Article

Journal of Fourier Analysis and Applications

, Volume 2, Issue 1, pp 29-48

First online:

Nonlinear Approximation by Trigonometric Sums

  • R.A. DeVoreAffiliated withDepartment of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • , V.N. TemlyakovAffiliated withDepartment of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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Abstract

We investigate the \(L_p\)-error of approximation to a function \(f\in L_p({\Bbb T}^d)\) by a linear combination \(\sum_{k}c_ke_k\) of \(n\) exponentials \(e_k(x):= e^{i\langle k,x\rangle}=e^{i(k_1x_1+\cdots+k_dx_d)}\) on \({\Bbb T}^d,\) where the frequencies \(k\in {\Bbb Z}^d\) are allowed to depend on \(f.\) We bound this error in terms of the smoothness and other properties of \(f\) and show that our bounds are best possible in the sense of approximation of certain classes of functions.