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uncertainty principle

Type of Math Object: 
Example
Major Section: 
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Mathematics Subject Classification

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There exists a very general formulation of the mathematical uncertainty theorem
in the frame of wavelet theory. It applies to any function f(t), not only to the
gaussian one.

Let F(a) be the Fourier transform of f(t) (a stands for "omega"). The uncertainty
theorem states that the "time-width" D_t(f) and the "spectrum-width" D_a(F) satisfy
the following inequality:

D_t(f)D_a(F) >= 1/2

Equality is reached for a gaussian function.

f(t) is any complex function of the real variable t, satisfying some conditions
detailed at the bottom.

By analogy with "classical" quantum theory, the width of a function f is defined
as follows:

Let d_t(f) be the distribution function |f(t)|^2/||f||, where ||f|| is the norm of
f, that is the integral between +- infinity of |f|^2. The integral of the
distribution between +- infinity is of course 1. Then t_0 is the "average" value
of t treated as a random variable with distribution d_t(f). The variance of t will
be the average value of |t-t_0|^2, and the width of f is defined as the square root
of this variance. Similarly for F(a). But, if f(t) is real, the average a_0 is 0.

Like in quantum theory, the proof relies on the Cauchy-Schwarz inequality.

f(t) and its Fourier transform F(a) must satisfy the following conditions:

1 - f(t) is in L_2
2 - tf(t) is in L_2
3 - f'(t) (the derivative) is in L_2
4 - F(a) is in L_2
5 - aF(a) is in L_2

These conditions are probably not independent. It seems to me that the second one
implies all the others; can you prove it, or give a counter-example?

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