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An Introduction to the Uncertainty Principle: Hardy’s by Sundaram Thangavelu

By Sundaram Thangavelu

In 1932 Norbert Wiener gave a sequence of lectures on Fourier research on the Univer­ sity of Cambridge. One results of Wiener's stopover at to Cambridge was once his recognized textual content The Fourier quintessential and sure of its purposes; one other was once a paper by means of G. H. Hardy within the 1933 Journalofthe London Mathematical Society. As Hardy says within the creation to this paper, This be aware originates from a comment of Prof. N. Wiener, to the influence that "a f and g [= j] can't either be very small". ... The theo­ pair of transforms rems which keep on with supply the main special interpretation attainable ofWiener's comment. Hardy's personal assertion of his effects, calmly paraphrased, is as follows, within which f is an integrable functionality at the genuine line and f is its Fourier rework: x 2 m If f and j are either zero (Ix1e- /2) for big x and a few m, then each one is a finite linear mixture ofHermite services. specifically, if f and j are x2 x 2 2 2 either O(e- / ), then f = j = Ae- / , the place A is a continuing; and if one x 2 2 is0(e- / ), then either are null.

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