Simultaneous use of discrete and continuous bases in quantum systems is not possible in the context of Hilbert spaces, but only in the more general structure of rigged Hilbert spaces (RHS). In addition, the relevant operators in RHS (but not in Hilbert space) are a realization of elements of a Lie enveloping algebra and support representations of semigroups. We explicitly construct here basis dependent RHS of the line and half-line and relate them to the universal enveloping algebras of the Weyl-Heisenberg algebra and su(1, 1), respectively. The complete sub-structure of both RHS and of the operators acting on them is obtained from their algebraic structures or from the related fractional Fourier transforms. This allows us to describe both quantum and signal processing states and their dynamics. Two relevant improvements are introduced: (i) new kinds of filters related to restrictions to subspaces and/or the elimination of high frequency fluctuations and (ii) an operatorial structure that, starting from fix objects, describes their time evolution.
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This means that any Cauchy sequence in Φ on the Hilbert space norm has a limit in .
For a thoroughly study of changes of representations involving continuous bases see Ref. 18.
Remember that we are considering antilinear mappings in our duals and not linear as customary.
In fact dμ(λ) = ρ(λ) dλ, where ρ(λ) is the Radom-Nikodym derivative of μ with respect to the Lebesgue measure. Then, we may define a new by without loss of generality.
For instance, , which has a meaning as a kernel.10
This converges in the strong operator sense on L2(ℝ).
