Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 18 / b / ii / Solution

The analogous orthogonal projection onto the cosine terms givesFor $n=0$, this means $a_0=L^{-1}{\int }_{-L}^{L}f(x)dx$; the actual constant in the Fourier series is $a_0/2$ because the constant function has squared norm $2L$.