Exercise 47, § 1

Let $X$ be a locally compact topological space and $\mu$ a probability measure on X. Let $(f_n)_{n\in\mathbf{N}^*}$ be an orthonormal family in $L^2(X, \mu)$. Let $(a_n)_{n\in\mathbf{N}^*}$ be a sequence of complex numbers such that the family $(a_n$ log($n$))$_{n\in\mathbf{N}^*}$ belongs to $\ell^2(\mathbf{N}^*)$.

For every finite set of integers I, we note

and we also write $s_n=s_{\{1,...,n\}}$.

a) Let $N\geqslant 1$. There exists a family $\mathscr{I}$ of non-empty intervals of $\{1, . . . ,N\}$ and a constant $C\geqslant 0$, independent of N, satisfying the following conditions:

(i) Every interval of $\{1, . . . ,N\}$ of the form $I =\{1, . . . ,M\}$ is a disjoint union of at most C log(M + 1) intervals in $\mathscr{I}$;

(ii) Every integer $n\in \{1, . . . ,N\}$ belongs to at most C log(N + 1) intervals in $\mathscr{I}$.

b) There exists a constant $C>0$ such that for every $N\in \mathbf{N}^*$, one has

$\int_X($sup$_{1\leqslant n\leqslant N}|s_n|^2)d\mu\leqslant C\sum_{n=1}^N|a_n|^2$.

(Let $\tau (x)$ be the smallest integer $\leqslant N$ such that sup$_{1\leqslant n\leqslant N}|s_n(x)|=|s_{\tau(x)}(x)|$, and let $S(x) =s_{\tau(x)}(x)$. Let $\mathscr{I}$ be a family of intervals as in the previous question; write

where $\mathscr{J}$ is a disjoint family of intervals in $\mathscr{I}$, then apply the Cauchy-Schwarz inequality.)

c) For $j\in \mathbf{N}^*$, put

The series of general term $(j+ 1)|S_j|^2$ converges $\mu$-almost everywhere in X. d) The sequence $(s_{2^j}(x))_{j\in\mathbf{N}^*}$ is a Cauchy sequence for $\mu$-almost every $x\in X$. Let $f(x)$ denote its limit, defined $\mu$-almost everywhere. e) The sequence $(s_n(x))_{n\in\mathbf{N}^*}$ converges to $f(x)$ for $\mu$-almost every $x\in X$.

f) Let $f\in L^1(\mathbf{T})$ such that

$\sum_{h\in\mathbf{Z}}$ log($|h|+ 1$)$^2|\widehat{f}(h)|^2<+\infty$.

For almost every $x$, the Fourier series of $f$ at $x$ converges symmetrically to $f(x)$ (“Rademacher-Menshov theorem”).