Week 4 Tutorial Solutions

Integrating

1. For every $n\in \mathbb{N}$ we have $|{\varphi }_n(x)|\le 1$ for all $x\in [0,1)$ so $$\int |{\varphi }_n{|}^2\mathsf{d}\mu \le \int 1\mathsf{d}\mu \le 2\pi$$ and ${\varphi }_n$ therefore belongs to ${\mathsf{L}}^2(X,\mathcal{B},\mu )$.
2. First we check that $⟨f,g⟩$ is well-defined. This follows from Young's inequality $$|\int f\cdot g\mathsf{d}\mu |\le \int |f\cdot g|\mathsf{d}\mu \le \int {\displaystyle \frac{|f{|}^2}{2}}+{\displaystyle \frac{|g{|}^2}{2}}\mathsf{d}\mu <\mathrm{\infty }$$ because $f$ and $g$ belong to ${\mathsf{L}}^2(X,\mathcal{B},\mu )$. It is bilinear because the integral is linear and certainly symmetric. Lastly we must check that $⟨f,f⟩=0$ implies $f=0$. This is indeed the case because $$0=\int |f{|}^2\mathsf{d}\mu =⟨f,f⟩$$ means exactly that $f=0$ in $L^2(X,\mathcal{B},\mu )$.
3. We need to verify that $⟨{\varphi }_n,{\varphi }_j⟩=0$ for all $n\ne j$ in $\mathbb{N}$ and that $⟨{\varphi }_n,{\varphi }_n⟩=1$ for all $n\in \mathbb{N}$. As all functions are continuous we can calculate the integral by calculating the Riemann integral of the product ${\varphi }_n{\varphi }_k$. This is then a calculus exercise.
4. We calculate that $$\begin{array}{rl}0\le & {\Vert f-\sum _{n=0}^N⟨f,{\varphi }_n⟩{\varphi }_n\Vert }_{\mathsf{2}}^{\mathsf{2}}\\ =& ⟨f-\sum _{n=0}^N⟨f,{\varphi }_n⟩{\varphi }_n,f-\sum _{k=0}^N⟨f,{\varphi }_k⟩{\varphi }_k⟩\\ =& ⟨f,f⟩-⟨f,\sum _{k=0}^N⟨f,{\varphi }_k⟩{\varphi }_k⟩-⟨\sum _{n=0}^N⟨f,{\varphi }_n⟩{\varphi }_n,f⟩\\ & +\sum _{n=0}^N\sum _{k=0}^N⟨f,{\varphi }_n⟩⟨f,{\varphi }_k⟩⟨{\varphi }_k,{\varphi }_n⟩\\ =& \Vert f{\Vert }_{\mathsf{2}}^2-\sum _{n=0}^N|⟨f,{\varphi }_n⟩{|}^2\end{array}$$ giving the desired inequality. Since $$\sum _{n=0}^N|(\mathcal{F}f)(n){|}^2\le \Vert f{\Vert }_{\mathsf{2}}^2$$ for all $N\in \mathbb{N}$ it must be the case that $$\sum _{n=0}^N|(\mathcal{F}f)(n){|}^2$$ converges to a finite value, which means $\mathcal{F}f$ belongs to ${\ell }^{\mathsf{2}}(\mathbb{N}\cup \{0\})$.