Babenko–Beckner inequality

Short description: Theorem of Fourier analysis

In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (qp)-norm of the n-dimensional Fourier transform is defined to be[1]

$\displaystyle{ \|\mathcal F\|_{q,p} = \sup_{f\in L^p(\mathbb R^n)} \frac{\|\mathcal Ff\|_q}{\|f\|_p},\text{ where }1 \lt p \le 2,\text{ and }\frac 1 p + \frac 1 q = 1. }$

In 1961, Babenko[2] found this norm for even integer values of q. Finally, in 1975, using Hermite functions as eigenfunctions of the Fourier transform, Beckner[3] proved that the value of this norm for all $\displaystyle{ q \ge 2 }$ is

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

Thus we have the Babenko–Beckner inequality that

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

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

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

Main ideas of proof

Throughout this sketch of a proof, let

$\displaystyle{ 1 \lt p \le 2, \quad \frac 1 p + \frac 1 q = 1, \quad \text{and} \quad \omega = \sqrt{1-p} = i\sqrt{p-1}. }$

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

The two-point lemma

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

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

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

A sequence of Bernoulli trials

The measure $\displaystyle{ d\nu }$ 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 $\displaystyle{ d\nu_n(x) }$ which is the n-fold convolution of $\displaystyle{ d\nu(\sqrt n x) }$ 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 $\displaystyle{ d\nu_n(x) }$ with respect to the elementary symmetric polynomials.

Convergence to standard normal distribution

The sequence $\displaystyle{ d\nu_n(x) }$ converges weakly to the standard normal 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 operator C above in terms of the elementary symmetric polynomials with respect to the measure $\displaystyle{ d\nu_n(x) }$ 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 (qp)-norm of the Fourier transform is obtained as a result after some renormalization.