Journal of Fourier Analysis and Applications

, Volume 2, Issue 1, pp 29–48

Nonlinear Approximation by Trigonometric Sums

  • R.A. DeVore
  • V.N. Temlyakov
Article

DOI: 10.1007/s00041-001-4021-8

Cite this article as:
DeVore, R. & Temlyakov, V. J Fourier Anal Appl (1995) 2: 29. doi:10.1007/s00041-001-4021-8

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.

Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • R.A. DeVore
    • 1
  • V.N. Temlyakov
    • 1
  1. 1.Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208USA