Wavelets of Y. Meyer. -- An optimal basis in \(C(0,1)\) (Q1324909)

Let \(k\) be a non-negative integer. The basis in the space \(C(0,1)\) of continuous 1-periodic functions is called \(k\)-optimal if for any \(m\in N\) and any \(f\in C(0,1)\), \[ \| f- S_ m(f)\|_ C\leq A_ k \omega_{k+1}\left(\textstyle{{1\over m}},f\right). \] Here \(S_ m(f)\) is the \(m\)-partial sum, \(\omega_ \ell(\delta,f)\) is the \(\ell\)th modulus of continuity. The author has proved that the system of Meyer wavelets is a \(k\)-optimal basis for all \(k\) simultaneously.