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

Off Functional Analysis

By Sundaram Thangavelu

In 1932 Norbert Wiener gave a chain of lectures on Fourier research on the Univer­ sity of Cambridge. One results of Wiener's stopover at to Cambridge used to be his recognized textual content The Fourier fundamental and likely of its purposes; one other used to be a paper by way of G. H. Hardy within the 1933 Journalofthe London Mathematical Society. As Hardy says within the creation to this paper, This observe originates from a comment of Prof. N. Wiener, to the impression that "a f and g [= j] can't either be very small". ... The theo­ pair of transforms rems which persist with supply the main specified interpretation attainable ofWiener's comment. Hardy's personal assertion of his effects, flippantly paraphrased, is as follows, during which f is an integrable functionality at the genuine line and f is its Fourier remodel: 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 blend ofHermite capabilities. particularly, 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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