On the Fourier coefficients of a discontinuous function

Abstract

We suppose throughout that f(t) is periodic with period 2π, and Lebesgue-integrable in (− π, π).We writeand suppose that the Fourier series of φ(t) and ψ(t) are respectively cos nt and sin nt. Then the Fourier series and allied series of f(t) at the point t = x are respectively and , where A0 = ½a0, An = ancos nx + bnsin nx, Bn = bncos nx − ansin nx and an, bn are the Fourier coefficients of f(t).