On the decay - and the smoothness behavior of the Fourier transform, and the construction of signals having strong divergent Shannon sampling series

Abstract

In this work, we show by means of the technique inspired by the Banach-Steinhaus Thm., that typically the Fourier transform of an integrable signal decays arbitrarily slowly toward the infinity, and has an arbitrary weak worst continuity/smoothness behaviour. However, the corresponding characterization can only be given weakly by means of the limit superior. Those statements gives therefore a tightening of the famous Riemann-Lebesgue's Lemma. Furthermore, we give a construction of functions, whose Fourier transform decays slowly than an arbitrary given decay rate. Inspired by that, we are also able to give an alternative proof of the strong divergence of the Shannon sampling series [1] for signals in the Paley-Wiener space PW(ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> , band-limited to an arbitrary ω <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">g</sub> ϵ ℝ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> . The corresponding construction of signals is stronger than the existent one given by Boche and Farell, and gives a new insight into the divergence phenomenon of the Shannon sampling series.