In mathematics, theBabenko–Beckner inequality (afterKonstantin I. Babenko [ru] andWilliam E. Beckner) is a sharpened form of theHausdorff–Young inequality having applications touncertainty principles in theFourier analysis ofL^p spaces. The(q, p)-norm of then-dimensionalFourier transform is defined to be^[1]

In 1961, Babenko^[2] found this norm foreven integer values ofq. Finally, in 1975, usingHermite functions aseigenfunctions of the Fourier transform, Beckner^[3] proved that the value of this norm for all${\displaystyle q\ge 2}$ is

${\displaystyle \Vert \mathcal{F}{\Vert }_{q,p}={\left(p^{1/p}/q^{1/q}\right)}^{n/2}.}$

Thus we have theBabenko–Beckner inequality that

${\displaystyle \Vert \mathcal{F}f{\Vert }_q\le {\left(p^{1/p}/q^{1/q}\right)}^{n/2}\Vert f{\Vert }_p.}$

To write this out explicitly, (in the case of one dimension,) if the Fourier transform is normalized so that

${\displaystyle g(y)\approx {\int }_{\mathbb{R}}{e}^{-2\pi ixy}f(x)dx\text{ and }f(x)\approx {\int }_{\mathbb{R}}{e}^{2\pi ixy}g(y)dy,}$

then we have

${\displaystyle {\left({\int }_{\mathbb{R}}|g(y){|}^{q}dy\right)}^{1/q}\le {\left(p^{1/p}/q^{1/q}\right)}^{1/2}{\left({\int }_{\mathbb{R}}|f(x){|}^{p}dx\right)}^{1/p}}$

or more simply

${\displaystyle {\left(\sqrt{q}{\int }_{\mathbb{R}}|g(y){|}^{q}dy\right)}^{1/q}\le {\left(\sqrt{p}{\int }_{\mathbb{R}}|f(x){|}^{p}dx\right)}^{1/p}.}$

Throughout this sketch of a proof, let

(Except forq, we will more or less follow the notation of Beckner.)

Let${\displaystyle d\nu (x)}$ be the discrete measure with weight${\displaystyle 1/2}$ at the points${\displaystyle x=\pm 1.}$ Then the operator

${\displaystyle C:a+bx\to a+\omega bx}$

maps${\displaystyle L^p(d\nu )}$ to${\displaystyle L^q(d\nu )}$ with norm 1; that is,

${\displaystyle {\left[\int |a+\omega bx{|}^{q}d\nu (x)\right]}^{1/q}\le {\left[\int |a+bx{|}^{p}d\nu (x)\right]}^{1/p},}$

or more explicitly,

${\displaystyle {\left[\frac{|a+\omega b{|}^q+|a-\omega b{|}^q}{2}\right]}^{1/q}\le {\left[\frac{|a+b{|}^p+|a-b{|}^p}{2}\right]}^{1/p}}$

for any complexa,b. (See Beckner's paper for the proof of his "two-point lemma".)

The measure${\displaystyle d\nu }$ that was introduced above is actually a fairBernoulli trial with mean 0 and variance 1. Consider the sum of a sequence ofn such Bernoulli trials, independent and normalized so that the standard deviation remains 1. We obtain the measure${\displaystyle d{\nu }_n(x)}$ which is then-fold convolution of${\displaystyle d\nu (\sqrt{n}x)}$ with itself. The next step is to extend the operatorC defined on the two-point space above to an operator defined on the (n + 1)-point space of${\displaystyle d{\nu }_n(x)}$ with respect to theelementary symmetric polynomials.

The sequence${\displaystyle d{\nu }_n(x)}$ converges weakly to the standardnormal probability distribution${\displaystyle d\mu (x)=\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx}$ with respect to functions of polynomial growth. In the limit, the extension of the operatorC above in terms of the elementary symmetric polynomials with respect to the measure${\displaystyle d{\nu }_n(x)}$ is expressed as an operatorT in terms of theHermite 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.

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