Reproducing kernel bounds for an advanced wavelet frame via the theta function

This work presents an analysis of the reproducing kernel K(p) associated to a class of wavelets K(p) derived from the theory of multiplicatively advanced differential equations. These kernels K(p) are expressible in terms of a family of new wavelets fk(t), which are generated by two fundamental wavelets and , all of which satisfy multiplicatively q-advanced perturbations of the second-order equation of the harmonic oscillator f″(t)=−f(t). As the parameter q→1+, and approach cos(t) and sin(t) uniformly, respectively, on compact subsets of R. Decay rates for K(p) as q→∞ are given, refining the understanding of the associated frame operator S, and providing for efficiency in inversion of S.