A new method for numerical evaluation of the Fourier transform

may encounter considerable difficulties due to the infinite interval of integration and the oscillatory integrand. In the absence of a universally accepted procedure for calculating j we are emboldened to suggest a new one. In what follows we simplify our work by imposing certain natural restrictions on the behavior of f andfi Thus (Aj) d eno es t an assumption about f and (&) denotes the same assumption about f. In Section 2 we give a simple and reasonable pair of restrictions on f sufficient to justify all the hypotheses of this section. For general Fourier integral background, see [2, 51. Denote by LP (p 2 l), as usual, the normed linear space of measurable functionsfon (---co, a) such that lifll; = syIf(x dx < co. L2 is a Hilbert space under the inner product (f / g) = l-“mf(~) g(x) dx. Assume first that f ELM n L2 (Al). Then 3(x) exists for all X, is continuous in x, vanishes at infinity, and 3~ L2. Moreover, the Fourier mapping F: f -3 of L1 n L2 into L2 is linear and [If/i, = jl f II2 . If we now assume that f is continuous (A2) and that ,f’eLl (A3), the inversion formula holds everywhere: