Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 7 / 2 / a

Use ${\mathbb{T}}^n={\mathbb{R}}^n/(2\pi \mathbb{Z}{)}^n$ and the periodic Sobolev space conventionThe Cauchy-Schwarz inequality givesTo check the last sum, partition the lattice into $|k|<1$ and dyadic shells $2^j\le |k|<2^{j+1}$. A shell contains at most $C_n{2}^{jn}$ points and contributes at most $C_n{2}^{j(n-2s)}$. This is a convergent geometric series exactly when $s>n/2$.

Thus the Fourier series ${\sum }_k\hat{f}(k)e^{ik\cdot x}$ converges absolutely and uniformly on the torus. Its sum is a continuous function and has the original Fourier coefficients, so it represents the original periodic distribution. In particular the representative is unique andThis proves the requested inclusion, with its continuous Sobolev embedding estimate.