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

The study of wavelets that satisfy the advanced differential equation K′(t)=K(qt) is continued. The connections linking the theories of theta functions, wavelets, and advanced differential equations are further explored. A direct algebraic–analytic estimate is given for the maximal allowable translation parameter N(q) such that b1 and any b>0 we find conditions guaranteeing that Λ(p,q,b)≡{(qm/2/‖K(p)‖)K(p)(qmt−nb)|m,n∈Z} is a wavelet frame for L2(R) where K(p) denotes the pth derivative/antiderivative of K. The frames Λ(p,q,b) become snug as either p→−∞ or q→∞, and their lower frame bounds A(p,q,b)→∞ as q→∞.