Applications of an advanced differential equation in the study of wavelets (Q1026459)

Let \(K\) be a function such that \(K(t)=0\) for \(t<0\) and \(K(t)=\sum_{k=-\infty}^{+\infty} (-1)^k \frac{e^{-q^k t}}{q^{k(k+1)/2}}\) for \(t\geq 0\). This function \(K\) is a solution of the multiplicatively advanced differential equation \(\frac{dK}{dt}(t)=K(qt)\). In this paper, the authors consider the following wavelet system \(\{ \psi_{m,n}(t):=\frac{q^{m/2}}{\sqrt{c_0}} K(q^mt-nb) : m,n\in \mathbb{Z}\}\), where \(c_0\) is a normalization constant. For each \(q>1\) and sufficient small \(b>0\), the authors show that this wavelet system is a frame of \(L^2(\mathbb{R})\). All the moments of \(K\) vanish and \(K\) belongs to the Schwarz class. Using the Jacobi theta function, the authors present an explicit estimation of the lower and upper frame bounds of the wavelet system in Proposition~2.