Extend Problem 28.9 to the Haar wavelet expansion.

[ use Problem 28.9 and show that \(\left\|\mathbb{E}^{\mathscr{A}_{-N}^{\Delta}} uight\|_{\infty} ightarrow 0\) for all \(u \in C_{c}(\mathbb{R})\).]

Data from problem 28.9

Show that the Haar-Fourier series for \(u \in C_{c}\) converges uniformly for every \(x\) to \(u(x)\). Show that this remains true for functions \(u \in C_{\infty}\), i.e. the set of continuous functions such that \(\lim _{|x| ightarrow \infty} u(x)=0\).

[ use the fact that \(u \in C_{c}\) is uniformly continuous. For \(u \in C_{\infty}\) observe that \(C_{\infty}=\bar{C}_{c}^{\|\cdot\|_{\infty}}\) (closure in sup-norm) and check that \(\left|s_{N}(u ; x)ight| \leqslant\|u\|_{\infty}\).]