Theta function identities in the study of wavelets satisfying advanced differential equations (Q982551)

The authors continue the study of the mother wavelet \[ K(t)=\sum_{k=-\infty}^{\infty}\,(-1)^k\,{e^{-q^k t}\over q^{k(k+1)/2}}\;\;(q>1;t\geq 0);\;\;K(t)=0\;\text{for }t\leq 0, \] that satisfies \[ K^{(p)}(t)=q^{p(p-1)/2}K(q^p t),\;K^{(p)}(t)=q^p K^{(p)}(qt) \] for all integers \(p\) (for \(p<0\) the antiderivative is used). \vskip0.2cm The main results concern condidiotns to ensure that the families \[ (q^{m/2}/\sqrt{c_0})K(q^m t-nb) \] and \[ (q^{m/2}/||K^{(p)}||)K^{(p)}(q^m t-nb) \] are waveletbases for \(L^2({\mathbb{R}})\). Moreover estimates are given and limit-properties for the qoutient of the upper and lower frame bounds for \(p\rightarrow -\infty,q\rightarrow\infty\). The proofs make extensive use of the Jacobi theta function.