Wavelet-type frames for an interval

Abstract

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λk,n(γ)=(1−γ2n)(ωn)k,n∈N, 0≤k<n, where ωn=e2πi/n is the primitive nth root of unity, and obtain a full description, in terms of the parameter γ, of the sets {λk,n(γ)} yielding frames for the space L2(−1,1). This is done using a disc version of the Bargmann transform, which maps L2(−1,1) to the classical spaces of the unit disc (Bergman, Hardy, Dirichlet) and applying Seip’s description of sampling sets in the unit disc. We also describe how to replace the points λk,n(γ) by the orbit of a Fuchsian group Γ, observing that Seip’s density of the orbit of a point contained in the fundamental region of a Fuchsian group Γ is equal to m0, where m0 is the smallest number among the weights of the automorphic forms with a zero in the fundamental region of the group Γ.