The remainder term in Fourier series and its relationship with the Basel problem

摘要

In this paper it is shown several approximation formulae for the remainder term of the Fourier series for a wide class of functions satisfying specific boundary conditions. Also it is shown that the remainder term is related with the Basel problem and the Riemann zeta function, which can be interpreted as t he energy of discrete-time signals; from this point of view, their energy can be calculated with a direct formula instead of an infinite series. The validity of this algorithm is established by means several proofs.