Asymptotic approximations by Harold Jeffreys

By Harold Jeffreys

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Gelfand (cf. Gelfand (1960) or, for example, Vilenkin (1965)) and is based on the representation of the n-dimensional torus ,][,n in the form ,][,n = JRn /lLn . The inversion formula for the Fourier integral is then reduced to the inversion formula for Fourier series. 28 Chapter 1. Convolution and Translation in Classical Analysis We consider for simplicity the case n j(A) = _1_ r f(A)e-i>"xdA = Vfff i'R,n = where = 1 . x. (6) 0 L L rk kEZ The series (6) converges absolutely and uniformly (even after differentiating 4> with respect to x and A).

Thus 'I/J E Hom(G, C*), and, if Hom(G, C*) is provided with the topology of uniform convergence on compacts in G, then Hom(G, C*) becomes a topological group. The group C* is isomorphic to the direct product lR* x 'IT' and therefore to lRx'IT'. Therefore, Hom(G, C*) is topologically isomorphic to the direct product G x Hom(G, lR). ) E G x Hom(G, lR) corresponds a generalized character 'I/J(X) = X (X)e"A (X) , XEG, (2) where >,(x) obeys >'(XI + X2) = >'(Xl) + >'(X2), XI,X2 E G. The elements of the group Hom(G, lR) are called the real characters of the group G.

Therefore, all the properties of Fourier transforms can be reformulated in the language of Mellin transforms. l,From our purposes here, the most important interpretation of the Mellin transform is the following. 2. In classical harmonic analysis one singles out yet another class of transforms, which are called the Hankel transforms, 1tv = 1t. r(1/ + k + 1)" 1 00 g(x) y'xYJv (xy)dx. Many facts of the Fourier transform theory, including the uncertainty principle, are applicable to Hankel transforms (for an example, see Papoulis (1968)) .

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