Past exam of the mathematical methods I course of the University of Oxford / 2015 resit / 12 / b / Solution

After discarding $f_0=0$, the preceding inner products show that distinct sine functions are orthogonal; dividing each by $\sqrt{\pi }$ makes them orthonormal. More precisely, their infinite expansions form an orthogonal basis of the odd part of $L^2(-\pi ,\pi )$ in the Hilbert-space sense. This is not a claim that every arbitrary odd function is a finite linear combination of sines.

Take the inner product of $x={\sum }_{n\ge 1}{a}_n\sin (nx)$ with $\sin (mx)$. Orthogonality givesTherefore the Fourier series of a periodic sawtooth, scaled to this interval, isIts periodic extension is piecewise smooth, so the Dirichlet convergence theorem for Fourier series gives pointwise convergence to $x$ inside the interval. At the jump endpoints of that periodic extension, the series instead converges to zero, the midpoint of the two one-sided limits.