Wavelet-type frames for an interval

摘要

We construct a sequence of rational functions, which will be called disc wavelets, parametrized by points in the unit disc λ_(k,n) (γ) = (1 - γ/2n)(ω_n)~k, n ∈ N, 0 ≤ k < n, where ωn = e~(2πi) is the primitive nth root of unity, and obtain a full description, in terms of the parameter γ, of the sets {λ_(k,n) (γ)} yielding frames for the space L2(-1, 1). This is done using a disc version of the Bargmann transform, which maps L~2(-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 λ_(k,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 m_0, where m_0 is the smallest number among the weights of the automorphic forms with a zero in the fundamental region of the group Γ.