Abstract
We study the Fourier transform windowed by a bump function. We transfer Jackson’s classical results on the convergence of the Fourier series of a periodic function to windowed series of a not necessarily periodic function. Numerical experiments illustrate the obtained theoretical results.
Highlights
The theory of Fourier series plays an essential role in numerous applications of contemporary mathematics
In the present paper we focus on the reconstruction of a not necessarily periodic function with respect to a finite interval (−λ, λ)
We investigate the convergence speed of Fourier series windowed by compactly supported bump functions with a plateau
Summary
The theory of Fourier series plays an essential role in numerous applications of contemporary mathematics. The Fourier coefficients of such windows may exceed spectral convergence (faster than any fixed polynomial rate), it is their compact support which limits the order to be at most root exponential and the actual convergence rate depends on the growth of the window’s derivatives, see e.g. In [3] a smooth bump is designed such that the order of the windowed Fourier coefficients is root-exponential (at least for the saw wave function), wheres in [24] we find a non-compactly supported window, for which we obtain true exponential decay. We investigate the convergence speed of Fourier series windowed by compactly supported bump functions with a plateau. In Theorem 4.6 we connect the decay rate of windowed Fourier coefficients to a new bound for the variation of windowed functions, which is based on the combination of two main ingredients: the Leibniz product rule and a bound for intermediate derivatives due to Ore
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