The single underlying method of “averaging the wavelet functional over translates” yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on the equivalence of affine and quasiaffine frames. The method applies to mu...

We characterize sequences of numbers (an) such that ∑ n≥1 anΦn converges a.e. for any orthonormal system (Φn) in any L2-space. In our criterion, we use the set B = { ∑ m≥n |am| ;n ≥ 1} and its information function hB(t) = − log3(β − α) for t ∈ (α, β], [α, β] ∩B = {α, β}.

Let {φn} be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions, the rational functions φn form an orthonormal system with respect to a Hermitian positive-definite inner product.

Let {φn} be a sequence of rational functions with arbitrary complex poles, generated by a certain three-term recurrence relation. In this paper we show that under some mild conditions the rational functions φn form an orthonormal system with respect to a Hermitian positive-definite inner product.

For an arbitrary full rank lattice Λ in R and a function g ∈ L(R) the Gabor (or Weyl-Heisenberg) system is G(Λ, g) := {eg(x − κ) ̨ ̨ (κ, `) ∈ Λ}. It is well-known that a necessary condition for G(Λ, g) to be an orthonormal basis for L(R) is that the density of Λ has D(Λ) = 1. However, except for symplectic lattices it remains an unsolved question whether D(Λ) = 1 is sufficient for the existence o...

We establish a one-to-one correspondence between (1) quantum teleportation schemes, (2) dense coding schemes, (3) orthonormal bases of maximally entangled vectors, (4) orthonormal bases of unitary operators with respect to the Hilbert-Schmidt scalar product, and (5) depolarizing operations, whose Kraus operators can be chosen to be unitary. The teleportation and dense coding schemes are assumed...