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 used to be his recognized textual content The Fourier essential and sure of its purposes; one other used to be a paper through G. H. Hardy within the 1933 Journalofthe London Mathematical Society. As Hardy says within the advent to this paper, This notice 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 stick to provide the main detailed interpretation attainable ofWiener's comment. Hardy's personal assertion of his effects, evenly 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 services. 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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