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The Fourier transform g (k) of a square integrable function f(x), vanishing for x < 0, is analytic in the upper half plane, so that, replacing k by k exp ζ, k≥0, 0 < Im ζ < π, it can be associated with an operator K(ζ) in H[sub +] = L ²((0, ∞),dx). The operator K(ζ) can be expressed in terms of the generator D of the dilatation group on H[sub +] and it can be shown that it is analytic in the strip 0 < Im ζ < π with strong limits as Im ζ↓0 and ↑r. The Laplace transform (ζ = iπ/2) is an analytic vector for D. It is also found that D is not a spectral operator of scalar type on L[sup p] ((0, ∞),dx), 1≤p < ∞, p ≠ 2. Applying the results obtained here to the time-evolution operator for a one-dimensional Sommerfeld model for the interaction between an electron and a metal, it is found that this operator has a complex-dilated analytic extension. [ABSTRACT FROM AUTHOR]