Babenko–Beckner inequality
Encyclopedia
In mathematics, the Babenko–Beckner inequality is a sharpened form of the Hausdorff–Young inequality
having applications to uncertainty principle
s in the Fourier analysis of Lp spaces
. The (q, p)-norm of the n-dimensional Fourier transform
is defined to be
In 1961, Babenko found this norm for even integer values of q. Finally, in 1975,
using Hermite functions as eigenfunction
s of the Fourier transform, Beckner
proved that the value of this norm for all
is
Thus we have the Babenko–Beckner inequality that
To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that
then we have
or more simply
(Except for q, we will more or less follow the notation of Beckner.)
be the discrete measure with weight 
at the points 
Then the operator
maps
to 
with norm 1; that is,
or more explicitly,
for any complex a, b. (See Beckner's paper for the proof of his "two-point lemma".)
that was introduced above is actually a fair Bernoulli trial
with mean 0 and variance 1. Consider the sum of a sequence of n such Bernoulli trials, independent and normalized so that the standard deviation remains 1. We obtain the measure
which is the n-fold convolution of 
with itself. The next step is to extend the operator C defined on the two-point space above to an operator defined on the (n+1)-point space of 
with respect to the elementary symmetric polynomials.
converges weakly to the standard normal probability distribution 
with respect to functions of polynomial growth. In the limit, the extension of the operator C above in terms of the elementary symmetric polynomials with respect to the measure 
is expressed as an operator T in terms of the Hermite polynomials
with respect to the standard normal distribution. These Hermite functions are the eigenfunctions of the Fourier transform, and the (q, p)-norm of the Fourier transform is obtained as a result after some renormalization.
Hausdorff–Young inequality
In mathematics, the Hausdorff−Young inequality bounds the Lq-norm of the Fourier coefficients of a periodic function for q ≥ 2. proved the inequality for some special values of q, and proved it in general...
having applications to uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
s in the Fourier analysis of Lp spaces
Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
. The (q, p)-norm of the n-dimensional Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
is defined to be
In 1961, Babenko found this norm for even integer values of q. Finally, in 1975,
using Hermite functions as eigenfunction
Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of the Fourier transform, Beckner
William E. Beckner
-External links:*...
proved that the value of this norm for all
isThus we have the Babenko–Beckner inequality that
To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that
then we have
or more simply
Main ideas of proof
Throughout this sketch of a proof, let(Except for q, we will more or less follow the notation of Beckner.)
The two-point lemma
Let be the discrete measure with weight at the points Then the operatormaps
to with norm 1; that is,or more explicitly,
for any complex a, b. (See Beckner's paper for the proof of his "two-point lemma".)
A sequence of Bernoulli trials
The measure that was introduced above is actually a fair Bernoulli trialBernoulli trial
In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, "success" and "failure"....
with mean 0 and variance 1. Consider the sum of a sequence of n such Bernoulli trials, independent and normalized so that the standard deviation remains 1. We obtain the measure
which is the n-fold convolution of with itself. The next step is to extend the operator C defined on the two-point space above to an operator defined on the (n+1)-point space of with respect to the elementary symmetric polynomials.Convergence to standard normal distribution
The sequence converges weakly to the standard normal probability distribution with respect to functions of polynomial growth. In the limit, the extension of the operator C above in terms of the elementary symmetric polynomials with respect to the measure is expressed as an operator T in terms of the Hermite polynomialsHermite polynomials
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence that arise in probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; and in physics, where...
with respect to the standard normal distribution. These Hermite functions are the eigenfunctions of the Fourier transform, and the (q, p)-norm of the Fourier transform is obtained as a result after some renormalization.