On Fourier coefficients of continuous functions for Haar-type systems (Q1311510)

Let \(\{\chi (p_ n)\}\) be a Haar-type system built from the sequence \(\{p_ n\}\) \((n \in \mathbb{N})\). Two theorems about the Fourier coefficients \(a_ n(f)\) of a continuous functions \(f\) for the system \(\{\chi (p_ n)\}\) are proved. For example, if \(f \in \text{Lip} \alpha\) \((0<\alpha \leq 1)\) on \([0,1]\) and \(\sum^ \infty_{n=1} | a_ n(f) |^{2 \alpha/(2 \alpha+1)}<\infty\), then \(f(t) \equiv \text{const.}\) on \([0,1]\).