Gaussian bases
Consider the functions \(\exp (-\zeta x^2)p_j(x)\) with \(p_j(x)= \sin [\pi/2 (x+1) (j+1/2)]\) and \(\text{Re} \zeta>0\). It is shown in this note that the family \(h_{j,k} (x)=h_j(x-2k)\), \(j=0,1, \dots, k\in Z\) is an unconditional basis for \(L^2(R)\). Also the dual basis for this is constructed. When \(\zeta>0\), this result is implicit in \textit{I. Daubechies}' ``Ten lectures on wavelets'' (1992; Zbl 0776.42018), page 120. It is more explicit on page 112, formula (28) in \textit{I. Daubechies} paper in \textit{D. H. Feng} et al. (eds.): ``Coherent states, past, present and future'', World Sci. Publ. 103-117 (1994)].
scientific article; zbMATH DE number 850662