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

Abstract

The study of wavelets that satisfy the advanced differential equation K ′ ( t ) = K ( q t ) 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 b < N ( q ) guarantees Λ ( 0 , q , b ) ≡ { ( q m / 2 / c 0 ) K ( q m t − n b ) | m , n ∈ Z } is a wavelet frame for L 2 ( R ) , where c 0 is the L 2 norm of K. For any q > 1 and any b > 0 we find conditions guaranteeing that Λ ( p , q , b ) ≡ { ( q m / 2 / ‖ K ( p ) ‖ ) K ( p ) ( q m t − n b ) | m , n ∈ Z } is a wavelet frame for L 2 ( 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 → ∞ .