We prove that, given $2< p<\infty$ , the Fourier coefficients of functions in $L_2(\mathbb {T}, |t|^{1-2/p}\,{\rm d}t)$ belong to $\ell _p$ and $1< p<2$ series sequences \vert {t}\vert ^{2/p-1}\,{\rm . Then, we apply these results study conditional Schauder bases almost greedy Banach spaces. Specifically, for every $0\le \alpha <1$ there is a basis whose conditionality constants ...