Hausdorff–Young inequality

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In mathematics, the Hausdorff−Young inequality bounds the L^q-norm of the Fourier coefficients of a periodic function for q ≥ 2. William Henry Young (1913) proved the inequality for some special values of q, and Hausdorff (1923) proved it in general. More generally the inequality also applies to the Fourier transform of a function on a locally compact group, such as Rⁿ, and in this case Babenko (1961) and Beckner (1975) gave a sharper form of it called the Babenko–Beckner inequality.

We consider the Fourier operator, namely let T be the operator that takes a function $f$ on the unit circle and outputs the sequence of its Fourier coefficients

$\widehat{f}(n)=\frac{1}{2\pi}\int_0^{2\pi}e^{-inx}f(x)\,dx,\quad n=0,\pm1,\pm2,\dots.$

Parseval's theorem shows that T is bounded from $L 2$ to $\ell^2$ with norm 1. On the other hand, clearly,

$|(Tf)(n)|=|\widehat{f}(n)|=\left|\frac{1}{2\pi}\int_0^{2\pi}e^{-int}f(t)\,dt\right|\leq \frac{1}{2\pi} \int_0^{2\pi}|f(t)|\,dt$

so T is bounded from $L 1$ to $\ell^\infty$ with norm 1. Therefore we may invoke the Riesz–Thorin theorem to get, for any 1 < p < 2 that T, as an operator from $L p$ to $\ell^q$ , is bounded with norm 1, where

$\frac{1}{p}+\frac{1}{q}=1.$

In a short formula, this says that

$\left(\sum_{n=-\infty}^\infty |\widehat{f}(n)|^q\right)^{1/q}\leq \left( \frac{1}{2\pi}\int_0^{2\pi}|f(t)|^p\,dt\right)^{1/p}.$

This is the well known Hausdorff–Young inequality. For p > 2 the natural extrapolation of this inequality fails, and the fact that a function belongs to $L p$ , does not give any additional information on the order of growth of its Fourier series beyond the fact that it is in $\ell^2$ .

[edit] References

Babenko, K. Ivan (1961), "An inequality in the theory of Fourier integrals", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya 25: 531–542, MR 0138939, ISSN 0373-2436 English transl., Amer. Math. Soc. Transl. (2) 44, pp. 115–128
Beckner, William (1975), "Inequalities in Fourier analysis", Annals of Mathematics. Second Series 102 (1): 159–182, MR 0385456, ISSN 0003-486X, http://www.jstor.org/stable/1970980
Hausdorff, Felix (1923), Eine Ausdehnung des Parsevalschen Satzes über Fourierreihen, 16, pp. 163–169, doi:10.1007/BF01175679
Young, W. H. (1913), "On the Determination of the Summability of a Function by Means of its Fourier Constants", Proc. London Math. Soc. 12: 71–88, doi:10.1112/plms/s2-12.1.7

Hausdorff–Young inequality

From Wikipedia, the free encyclopedia

[edit] References

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