2. VARIETY OF MATHEMATICAL FORMS OF UP 5
This interpretation remains meaningful even in certain
non-L2
cases. For in-
stance, when f(x) =
eiλx
is a single-moded wave (harmonic) with the wave number
λ R, then
|f(x)|2dx
is the infinite Lebesgue measure but the momentum of the
wave can be calculated as p = λ. The Fourier transform
ˆ,
f understood in the
sense of distributions, is a point-mass placed at p, which reflects the fact that in
this particular situation the moment of the particle can be calculated precisely.
In the
L2-case,
f is an infinite combination of harmonics and its portrait in the
momentum space cannot be localized. The classical Heisenberg’s inequality [148]
expresses this fact in terms of the variance of the measures
|f(x)|2dx
and |
ˆ(x)|2dx.
f
If μ is a probability measure on R we define its variance V (μ) as
inf
a∈R
(x
a)2dμ(x)
with an agreement that for absolutely continuous measures μ = ρ(x)dx we write
V (ρ) instead of V (μ).
Heisenberg’s inequality says that if f
L2(R)
then
(1.2) V
(|f|2)V
(|
ˆ|2)
f
||f||2
4
16π2
.
A simpler (but easily equivalent) version of the same inequality estimates the
product of second moments of the measures:
(1.3)
R
x2|f(x)|2
R
x2|
ˆ(x)|2dx
f
||f||2
4
16π2
.
The equation occurs when f is a Gaussian. The proof is straight-forward via inte-
gration by parts, Parseval’s theorem and the Cauchy-Schwarz inequality.
A stronger inequality was conjectured by Hirschman [67] and Everett [47] in
1957 and proved by Beckner [9] in 1975. If μ, dμ(x) = ρ(x)dx, is a probability
measure, its Shannon entropy H(μ) (H(ρ)) is defined as
H(ρ) =
R
ρ(x) log ρ(x)dx.
If f L2(R), ||f||2 = 1 then the Hirschman uncertainty of f, defined as H(|f|2) +
H(|
ˆ|2),
f is non-negative. Moreover, it has a lower bound
H(|f|2)
+ H(|
ˆ|2)
f log
e
2
with the equality, once again, achieved when f is a Gaussian.
Many further variations and generalizations of Heisenberg’s inequality can be
found in mathematical literature, see [50] for a detailed discussion and further
references.
An integral inequality is not the only way to express UP in the settings of
Harmonic Analysis. As was mentioned in the introduction, the following more
abstract statement is universally considered to be its proper formulation: f and
ˆ
f cannot be simultaneously small.
The ‘smallness’ in this statement can be understood in a variety of ways, each
producing its own problem. If this statement is applied to an L2-function, the norm
must of course be fixed beforehand, as was done in our discussion above.
Heisenberg’s inequality interprets smallness in the sense of the second moment,
while Hirschman’s inequality measures smallness with entropy. Many other natural
meanings of ‘smallness’ of f and
ˆ
f have been considered by mathematicians, some
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