An Introduction to the Uncertainty Principle : Hardy's Theorem on Lie Groups PDF
by Sundaram Thangavelu
Part of the Progress in Mathematics series
Description
In 1932 Norbert Wiener gave a series of lectures on Fourier analysis at the Univer- sity of Cambridge.
One result of Wiener's visit to Cambridge was his well-known text The Fourier Integral and Certain of its Applications; another was a paper by G.
H. Hardy in the 1933 Journalofthe London Mathematical Society.
As Hardy says in the introduction to this paper, This note originates from a remark of Prof.
N. Wiener, to the effect that "a f and g [= j] cannot both be very small". ... The theo- pair of transforms rems which follow give the most precise interpretation possible ofWiener's remark.
Hardy's own statement of his results, lightly paraphrased, is as follows, in which f is an integrable function on the real line and f is its Fourier transform: x 2 m If f and j are both 0 (Ix1e- /2) for large x and some m, then each is a finite linear combination ofHermite functions.
In particular, if f and j are x2 x 2 2 2 both O(e- / ), then f = j = Ae- / , where A is a constant; and if one x 2 2 is0(e- / ), then both are null.
Information
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Download - Immediately Available
- Format:PDF
- Publisher:Birkhauser Boston
- Publication Date:06/12/2012
- Category:
- ISBN:9780817681647
Information
-
Download - Immediately Available
- Format:PDF
- Publisher:Birkhauser Boston
- Publication Date:06/12/2012
- Category:
- ISBN:9780817681647
