Commutative Harmonic Analysis II: Group Methods in by V.P. Havin, N.K. Nikolski, D. Dynin, S. Dynin, V.P. Gurarii

By V.P. Havin, N.K. Nikolski, D. Dynin, S. Dynin, V.P. Gurarii

Classical harmonic research is a vital a part of smooth physics and arithmetic, related in its value with calculus. Created within the 18th and nineteenth centuries as a unique mathematical self-discipline it persevered to increase (and nonetheless does), conquering new unforeseen components and generating outstanding purposes to a large number of difficulties, previous and new, starting from mathematics to optics, from geometry to quantum mechanics, let alone research and differential equations. the ability of workforce theoretic ideology is effectively illustrated by means of this wide selection of subject matters. it truly is commonly understood now that the reason of this superb energy stems from staff theoretic principles underlying essentially every little thing in harmonic research. This quantity is an strange mix of the final and summary staff theoretic procedure with a wealth of very concrete subject matters appealing to each person drawn to arithmetic. Mathematical literature on harmonic research abounds in books of roughly summary or concrete type, however the fortunate mix as within the current quantity can hardly ever be present in any monograph. This publication should be very priceless to a large circle of readers, together with mathematicians, theoretical physicists and engineers.

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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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