Babenko–Beckner inequality

In mathematics, the Babenko–Beckner inequality (after Konstantin I. Babenko [ru] and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of L^p spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be^[1]

${\displaystyle \|{\mathcal {F}}\|_{q,p}=\sup _{f\in L^{p}(\mathbb {R} ^{n})}{\frac {\|{\mathcal {F}}f\|_{q}}{\|f\|_{p}}},{\text{ where }}1<p\leq 2,{\text{ and }}{\frac {1}{p}}+{\frac {1}{q}}=1.}$

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 ${\displaystyle q\geq 2}$ is

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

Thus we have the Babenko–Beckner inequality that

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

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

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

or more simply

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

Main ideas of proof

Throughout this sketch of a proof, let

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

(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,

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

or more explicitly,

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

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 (q, p)-norm of the Fourier transform is obtained as a result after some renormalization.

See also

• Entropic uncertainty

References

• v
• t
• e

Lp spaces

Basic concepts

• Banach & Hilbert spaces
• L^p spaces
• Measure
  • Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces

L¹ spaces

• Integrable function
• Lebesgue integration
• Taxicab geometry

L² spaces

• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function

${\displaystyle L^{\infty }}$ spaces

• Bounded function
• Chebyshev distance
• Infimum and supremum
  • Essential
• Uniform norm

Maps

• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement

Inequalities

• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution

Results

• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem

For Lebesgue measure	Isoperimetric inequality Brunn–Minkowski theorem Milman's reverse Minkowski–Steiner formula Prékopa–Leindler inequality Vitale's random Brunn–Minkowski inequality

Applications & related

• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
  • C*-algebra
  • Von Neumann