| Topic: |
Science > Physics |
| User: |
"Vista" |
| Date: |
01 Jul 2007 07:31:09 PM |
| Object: |
looking for a theorem for Fourier transforms |
Hi all,
I am looking for a theorem relating the rate of decay of function f(x) for
large x, to the rate of decay of its Fourier Transform F(w) for large w.
I vaguely remember that a function f(x) with decay rate faster than
exp(-a*x) (a>0) for large x will have a Fourier Transform F(w) with decay
rate slower than exponential rate for large w.
Does anybody recall such a theorem? Could anybody give me some pointers to
papers anb books and literature..?
Thanks!
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| User: "" |
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| Title: Re: looking for a theorem for Fourier transforms |
01 Jul 2007 08:03:32 PM |
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On Jul 1, 5:31 pm, "Vista" <a...@gmai.com> wrote:
Hi all,
I am looking for a theorem relating the rate of decay of function f(x) for
large x, to the rate of decay of its Fourier Transform F(w) for large w.
I vaguely remember that a function f(x) with decay rate faster than
exp(-a*x) (a>0) for large x will have a Fourier Transform F(w) with decay
rate slower than exponential rate for large w.
This is not true. Indeed, a Gaussian like f(x)=e^{-x^2} is its own
Fourier transform (up to some constants) and decays faster than every
exponential.
At the other extreme, if f is something like a Dirac delta, then f is
identically 0 away from x=0 but the Fourier transform is e^{iw} which
does not even vanish at infinity.
In general, decay of the Fourier transform of f is usually related to
the *smoothness* of f, and vice versa. For example, the Fourier
transform of an L^1 function is continuous and vanishes at infinity
(but perhaps very slowly). If f is in the "Schwartz class" of
functions, so that f is smooth, and f and all its derivatives decay
faster than any rational function, then so is its Fourier transform.
These should be proved in any real analysis text.
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| User: "rge11x" |
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| Title: Re: looking for a theorem for Fourier transforms |
03 Jul 2007 11:26:00 AM |
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On Jul 1, 8:31 pm, "Vista" <a...@gmai.com> wrote:
Hi all,
I am looking for a theorem relating the rate of decay of function f(x) for
large x, to the rate of decay of its Fourier Transform F(w) for large w.
I vaguely remember that a function f(x) with decay rate faster than
exp(-a*x) (a>0) for large x will have a Fourier Transform F(w) with decay
rate slower than exponential rate for large w.
Does anybody recall such a theorem? Could anybody give me some pointers to
papers anb books and literature..?
Thanks!
I think you are referring to a theorem of Hardy:
Let F(y) = int_(-inf)^+inf){ f(x)exp(-i.2pi.x.y)dx} be the Fourier
transform of f(x) and assume that that there are positive constants a,
b, C >0 such that for all -inf <x,y<+inf:
|f(x)| < C exp(-pi.a.x^2) and
|F(y)| < C exp(-pi.b.y^2)
Then there are three cases
1. if ab>1 then f(x) = F(y) = 0 (almost) everywhere
2. if ab< 1 then there are infinitely many linearly independent
functions f(x) satisfying these inequalities.
3. if ab=1 then f(x) = exp(-pi.a.x^2)
This result, which you can find in Dym and McKean, has several
generalizations for other exponents, such as by Cowling and Price and
Morgan:
M.G. Cowling and J. F. Price, 'Generalizations of Heisenberg's
inequality', in: Harmonic analysis (eds. G. Mauceri, F. Ricci and G.
Weiss), LNM, no.992 (Springer, Berlin, 1983) pp. 443-449.
G.W. Morgan, 'A note on Fourier transforms', J. London Math. Soc. 9
(1934), 187-192.
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