Inequality for the first Fourier level of a Boolean function - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T23:56:19Z http://mathoverflow.net/feeds/question/89148 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/89148/inequality-for-the-first-fourier-level-of-a-boolean-function Inequality for the first Fourier level of a Boolean function Michal Kotowski 2012-02-22T00:12:09Z 2012-09-08T13:24:59Z <p>In the study of Boolean functions, the hypercontractive inequality enables one to bound from above the norm of $Tf$ by some norm of $f$, where $T$ is the noise operator depending on the noise parameter. This can be written in terms of the Fourier transform of $f$ as (in a special case of $q=2$):</p> <p>$\left( \sum\limits_{S \subseteq [n]} (p - 1)^{|S|} \widehat{f}(S)^2 \right)^{1/2} \leq \left( \frac{1}{2^n} \sum\limits_{x \in (0,1)^n} |f(x)|^p \right)^{1/p}$</p> <p>However, this involves all Fourier levels of $f$. In the application I have in mind, I'm interested only in bounding the norm of the first level of $f$, i.e. restricting the sum on the left to $|S| = 1$. Is it possible to give any inequality of this kind, probably with a different right hand side (but still involving some information about the norm of $f$) and some additional assumptions on $f$? If it's impossible for some trivial reasons, let me know anyway.</p> <p>Here I'd be mostly interested in matrix-valued Boolean functions (see for example <a href="http://arxiv.org/abs/0705.3806" rel="nofollow">http://arxiv.org/abs/0705.3806</a>), although any answer would be appreciated.</p> http://mathoverflow.net/questions/89148/inequality-for-the-first-fourier-level-of-a-boolean-function/106670#106670 Answer by Ryan O'Donnell for Inequality for the first Fourier level of a Boolean function Ryan O'Donnell 2012-09-08T13:24:59Z 2012-09-08T13:24:59Z <p>Here is some general discussion of the level-1 weight of Boolean functions --</p> <p><a href="http://analysisofbooleanfunctions.org/?p=885" rel="nofollow">http://analysisofbooleanfunctions.org/?p=885</a></p> <p>-- including the theorems that:</p> <ol> <li>If f is boolean-valued and all of its degree-1 coefficients are small then the weight at level 1 is not more than $2/\pi$;</li> <li>If f is boolean-valued and has very small variance $\alpha$ then the weight at level 1 is at most $2 \alpha \log_2(1/\alpha)$. ["Chang's Lemma", or "Talagrand's Lemma"]</li> </ol> <p>For functions that are not boolean-valued, I don't have a lot to say; the main thing I can suggest is taking $p$ in the hypercontractive inequality as you stated it very close to $1$; if it is, say, $1+\epsilon$ then the LHS will have $\widehat{f}(\emptyset)^2$ (which usually you have information about), plus $\epsilon$ times the weight at level 1, plus at most $\epsilon^2$ times the $2$-norm (neglectable if $\epsilon$ is small enough). So this may allow you to "isolate" the level-1 weight after subtracting $\widehat{f}(\emptyset)^2$ and dividing by $\epsilon$.</p>