 Half-range Fourier cosine series

For a function prescribed on $0<x<L$, its half-range Fourier cosine series is the Fourier series of the even extension, made $2L$-periodic. Its coefficients are $a_n=(2/L){\int }_0^{L}f(x)\cos (n\pi x/L)dx$, including $n=0$. At a jump the usual piecewise-smooth convergence theorem gives the average of the two one-sided limits of the periodic extension.