A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Question

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ converges pointwise almost everywhere on $[0,1]$?

I guess that this is not true. However, it does hold in some interesting special cases:

For example, if $f$ is $1$ on $[2k,2k+1)$ and $-1$ on $[2k-1,2k)$ for $k \in \mathbb Z$ and one defines $f_n(x) = f(2^nx)$, this is true (one proof relies on a theorem about almost sure convergence of independent random variables, see http://en.wikipedia.org/wiki/Kolmogorov%27s_three-series_theorem).

Another interesting case is the choice $f_n(x) = e^{2\pi i n x}$ where it is true by Carleson's theorem about the almost everywhere convergence of a Fourier series.

Answer 1

As you remark, this is false in the context of general orthonormal sequence. Generally one has the estimate (the Rademacher-Menshov theorem)

$$|| \max_{\ell \leq n} |\sum_{i=1}^{\ell} a_i \phi_i| ||_{L^2} \lesssim \log n (\sum_{i=1}^{n} |a_i|^2)^{1/2}$$

where, in general, the factor of $\log n$ is optimal.

Even more surprisingly, perhaps, is that for any complete orthonromal sequence there exists a reordering $\pi$ such that the orthonormal sequence $\{f_{\pi(n)} \}$ fails to have the almost everywhere convergence property. This is a theorem of Olveskii (see theorem 2 in Olveskii's book).