Generalized eigenfunctions may be regarded as vectors of a basis in a particular direct integral of Hilbert spaces or as elements of the antidual space Φ × in a convenient Gelfand triplet Φ ⊆ H ⊆ Φ × . This work presents a fit treatment for computational purposes of transformations formulas relating different generalized bases of eigenfunctions in both frameworks direct integrals and Gelfand triplets. Transformation formulas look like usual in Physics literature, as limits of integral functionals but with well defined kernels. Several approaches are feasible. Here Vitali and martingale approaches are showed.

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Spectral representations of the dilation and translation operators on L 2 (R) are built through appropriate bases. Orthonormal wavelets and multiresolution analysis are then described in terms of rigid operator-valued functions defined on the functional spectral spaces. The approach is useful for computational purposes.

Spectral decompositions of translation and dilation operators are built in terms of suitable orthonormal bases of L2(ℝ), leading to spectral formulas for scaling functions and orthonormal wavelets associated with multiresolution analysis (MRA). The spectral formulas are useful to compute compactly supported scaling functions and wavelets. It is illustrated with a particular choice of the orthonormal bases, the so-called Haar bases, which yield a new algorithm related to the infinite product matrix representation of Daubechies and Lagarias.

Motivated by physical problems, Misra, Prigogine and Courbage (MPC) studied the following problem: given a one-parameter unitary group {Ut} on a separable Hilbert space K, find a Hilbert space H, a contraction semigroup {Wt} on H and an injective operator Λ : H → K with dense range which intertwines the actions of {Ut} and {Wt} (ΛWt = UtΛ). More precisely, they studied the case where H = K is an L 2 -space over a probability space and both {Ut} and {Wt} are Markovian (i.e. positivity and identity preserving). MPC gave a sufficient condition for the existence of a solution of the above problem, the existence of a time operator associated to {Ut}. In this paper we prove that, using the Sz.-Nagy-Foiaş dilation theory, it is possible to give a constructive characterization of all the solutions of the MPC problem in the general context. This criterium allows one to construct a solution of the MPC problem for which no time operator exists. When specialized to L 2 -spaces and Markovian {Ut} and {Wt}, the present criterium is applied to address the so-called inverse problem of Statistical Mechanics, namely to characterize the intrinsically random dynamics {Ut}.