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.
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DeVore, R., Temlyakov, V. Nonlinear Approximation by Trigonometric Sums. J Fourier Anal Appl 2, 29–48 (1995). https://doi.org/10.1007/s00041-001-4021-8
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DOI: https://doi.org/10.1007/s00041-001-4021-8