The zero-one law for a complete orthonormal system (Q1886966)

Two theorems are announced without proofs. Theorem 1. There exists a complete orthonormal system (ONS) \(\{f_n: n= 1,2,\dots\}\), \(f_n\in L^\infty[0,1]\), such that the orthogonal series \(\sum a_nf_n(x)\) converges a.e. on \([0, 1]\) whenever \(\sum a^2_n< \infty\), and it diverges a.e. on \([0, 1]\) whenever \(\sum a^2_n= \infty\). Theorem 2. Let \(\{f_n\}\) be an arbitrary complete ONS on \([0, 1]\). Then there exists a fixed increasing sequence \(\{n_k\}\) of natural numbers with the following property: for every function \(f\) measurable and finite a.e. on \([0, 1]\) there exists a sequence \(\{b_n\}\) of coefficients such that \[ \lim_{k\to\infty} \sum^{n_k}_{n=1} b_nf_n(x)= f(x)\quad\text{a.e. on }[0,1]. \]