On Fourier series for higher order (partial) derivatives of functions

Abstract

This paper is focused on higher order differentiation of Fourier series of functions. By means of Stokes's transformation, the recursion relations between the Fourier coefficients in Fourier series of different order (partial) derivatives of the functions as well as the general formulas for Fourier series of higher order (partial) derivatives of the functions are acquired. And then, the sufficient conditions for term‐by‐term differentiation of Fourier series of the functions are presented. These findings are subsequently used to reinvestigate the Fourier series methods for linear elasto‐dynamical systems. The results given in this paper on the constituent elements, together with their combinatorial modes and numbering, of the sets of coefficients concerning 2rth order linear differential equation with constant coefficients are found to be different from the results deduced by Chaudhuri back in 2002. And it is also shown that the displacement solution proposed by Li in 2009 is valid only when the second order mixed partial derivative of the displacement vanishes at all of the four corners of the rectangular plate. Copyright © 2016 John Wiley & Sons, Ltd.

A novel method for high-frequency non-sinusoidal vibration waveforms with uniaxial electro-hydraulic shaking table based on Fourier series

“…In mathematics, the Fourier series is a way to represent a function as the sum of simple sine waves (Gogoladze and Tsagareishvili, 2016; Raj and Sharma, 2016; Rergis et al, 2018). It decomposes any periodic function or periodic signal into the sum of harmonically related sinusoidal functions (Rouba et al, 2018; Sun and Zhang, 2017; Telyakovskii, 2018). Similarly, a non-sinusoidal vibration waveform can be represented as the sum of sinusoidal vibration waveforms.…”