Salem Inequality - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T14:47:49Z http://mathoverflow.net/feeds/question/28295 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28295/salem-inequality Salem Inequality Vagabond 2010-06-15T18:04:07Z 2010-06-16T05:53:12Z <p>I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" <a href="http://www.math.msu.edu/~fedja/Published/paper.ps" rel="nofollow">http://www.math.msu.edu/~fedja/Published/paper.ps</a> by Nazarov and he calls it by the name of Salem Inequality (which according to him is well known but I cant find a reference).</p> <p>If I have understood it correctly the Inequality says that if $p$ is an exponential polynomial whose exponents are well separated, then the average value of square of the modulus of $p$ over a sufficiently large interval dominates the sum of the square of the modulus of its coefficients.</p> <p>Let $p(t) = \Sigma_{k=1}^n c_k e^{ i \lambda_k t}$, where $\lambda_1&lt;\lambda_2\dots&lt;\lambda_n \in \mathbb R$ and $\lambda_k$'s satisfies a separation condition i.e., $\lambda_{k+1}-\lambda_k \geq \Delta >0$. Let $I$ be an interval of length bigger than $4\pi / \Delta$, then $$\sum_{k=1}^{n} |c_k|^2 \leq \frac{4}{|I|} \int_I |p(t)|^2 dt.$$ How can one prove this Inequality? This surely would have a lot of appliction (and as he says must be well known !! may be by a different name ?). I would appreciate some references to such inequalities in general. Also I find curious that the length of the interval does not seem to depend on $n$ and depends only on $\Delta$.</p> http://mathoverflow.net/questions/28295/salem-inequality/28353#28353 Answer by Victor Protsak for Salem Inequality Victor Protsak 2010-06-16T05:53:12Z 2010-06-16T05:53:12Z <p>Following a cue from Wadim, this inequality is Theorem 9.1 in Chapter 5 of Zygmund's <em>Trigonometric series</em>, vol 1. Note that although the book is mostly dealing with trigonometric series, the proof is given for general lacunary $\lambda_k.$ (Salem was a good friend of Zygmund's; see the preface to the book.)</p>