The Fourier–Plancherel transform as a spectral representation of a rigged Hilbert space

Abstract

In ‘Generalized Translation Operators…’ [3] algebras associated with a self-adjoint operator were investigated. Examples were given in the cases, inter alia, of the operators Mt and i d/dt in the space L2(ℝ, m) (m = Lebesgue measure). This paper shows that by suitably rigging the space the examples can be seen as natural generalizations of certain familiar algebras. The rigging enables us to introduce, rigorously, into L2(ℝ,m) the improper elements used by Akhiezer and Glazman [1] as cyclic vectors for Mt and i d/dt in order to identify the Fourier–Plancherel transform with a spectral representation of the space.