Exercise 15, § 6

Let X be a compact space, $\mu$ a positive measure on X. For two functions $f, g$ of $\mathscr{L}_C^2$, one sets $(f|g) = (\widetilde{f}|\widetilde{g}) = \int f \overline{g} d\mu$. A sequence of functions $f_n \in \mathscr{L}_C^2$ is said to be orthonormal if the sequence $\widetilde{f}_n$ is orthonormal in the Hilbert space $L_C^2$, that is (TVS, V, §2) if $(f_m|f_n) = \delta_{mn}$ (the Kronecker symbol) for every pair of indices. For every function $g \in \mathscr{L}_C^2$, the complex numbers $c_n = (g|f_n)$ are called the components of $g$ with respect to the orthonormal sequence $(f_n)$; one has $\sum_{n=0}^\infty |c_n|^2 \leq \int |g|^2 d\mu$.

For every pair of points $x, y$ of X and every integer $n \geq 0$, one sets $K_n(x, y) = \sum_{k=0}^n f_k(x) \overline{f_k(y)}$ (the n-th kernel of the orthonormal sequence $(f_n)$); for every function $g \in \mathscr{L}_C^2$,

Set $H_n(x) = \int |K_n(x, y)|\,d\mu(y)$ (called the n-th Lebesgue function of the orthonormal sequence $(f_n)$).

a) Let $(\alpha_n)$ be a decreasing sequence of numbers $> 0$ such that the series with general term $\alpha_n$ is convergent. Show that, for almost every $x \in X$, $\sum_{k=0}^n |f_k(x)|^2 = o(1/\alpha_n)$ (using Prop. 6 of §3, show that the series with general term $\alpha_n|f_n(x)|^2$ is convergent almost everywhere, and make use of Exer. 10 of GT, IV, §7). Deduce from this that $H_n(x) = o(1/\sqrt{\alpha_n})$ for almost every $x \in X$.

b) Let $x_0$ be a point of X. In order that, for every complex-valued function $g$, defined and continuous on X, the partial sums $s_n(g)$ be bounded at the point $x_0$ (by a number depending on $g$ and $x_0$), it is necessary and sufficient that the set of numbers $H_n(x_0)$ be bounded (make use of Prop. 3 and the fact that in the dual of a Banach space, every weakly bounded set is strongly bounded).

c) In order that, for every complex-valued function $g$, defined and continuous on X, the series with general term $(g|f_n)f_n(x)$ be uniformly convergent in X and have sum $g(x)$, it is necessary and sufficient that: 1° every continuous complex-valued function on X be uniformly approximable by linear combinations of the $f_k$; 2° there exist a constant $a$ such that $|H_n(x)| \leq a$ for all $n$ and $x \in X$. (Note that for every $n$, $f_n(x) = \sum_{m=0}^\infty (f_n|f_m)f_m(x)$ identically; on the other hand, to prove the necessity of the condition $2^\circ$, note that for every increasing sequence $(n_k)$ of integers and every sequence $(x_k)$ of points of $X$, the sequence of numbers $\int K_{n_k}(x_k, y)g(y)\, d\mu(y)$ is bounded (by a number depending on $g$) and argue as in $b$.)