Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 18

A standard complete orthogonal family is the real Fourier basisCompleteness gives expansion in the L2 space for a square-integrable function. Mere integrability alone does not guarantee pointwise convergence of a Fourier series; the explicit function in part (b) is continuous and piecewise smooth, so this issue does not obstruct its expansion.

With inner product $⟨u,v⟩={\int }_{-\pi }^{\pi }u(x)v(x)dx$, orthogonality meansThe constant function has squared norm $2\pi$, and each sine or cosine has squared norm $\pi$. Thus the displayed family is orthogonal but not normalized; these different lengths determine the coefficient factors.

The standard Fourier series iswith Fourier coefficientsThese follow by taking inner products with the individual basis vectors and using their orthogonality and squared norms.

Separate the already finite sine terms from the even part $g(x)=x^2-\pi |x|$. Since $g$ is an even function, it has no sine coefficients. Its constant coefficient isFor $n\ge 1$, integration by parts givesTherefore the cosine Fourier coefficients areOnly even frequencies survive: $a_{2m}=1/m^2$. The sine Fourier coefficients are $b_1=1$, $b_2=-2$, with all others zero. HenceThe periodic function is continuous at zero and at its endpoint joins, and its coefficients are absolutely summable. Thus this Fourier series represents it everywhere, including those joins.

Evaluate at $x=\pi /2$, where the original expression gives $f(\pi /2)=1-{\pi }^2/4$. The Fourier series givesRearranging provesThe convergence at this point is justified by the continuity properties noted in part (b).