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I have come across this inequality in the paper "Local estimates for exponential polynomials and their applications to inequalities of the uncertainty principle type" http://www.math.msu.edu/~fedja/Published/paper.ps by Nazarov and he calls it by the name of Salem Inequality (which according to him is well known, but I can't find a reference).

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.

Let $p(t) = \Sigma_{k=1}^n c_k e^{ i \lambda_k t}$, where $ \lambda_1<\lambda_2\dots<\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 applications (and as he says must be well known – maybe under a different name?). I would appreciate some references to such inequalities in general. Also I find it curious that the length of the interval does not seem to depend on $n$ and depends only on $\Delta$.

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    $\begingroup$ In Montgomery's book "Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis" the analogue for the maximum norm is stated on page 89. $\endgroup$
    – Helge
    Jun 15, 2010 at 18:41
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    $\begingroup$ The result you just mentioned is precisely what Nazarov uses to obtain the analogus result to arbitrary measurable sets (he calls it Turan Lemma). Its at this point he mentions Salem Inequality : I quote "Often it is desirable to have an upper estimate of the sum of absolute values of coefficients $\Sigma_{k=1}^n |c_k|$ rather than of the maximum $\max_{t\in I} |p(t)|$.... a desired estimate can be derived by using the well known Salem Inequality. $\endgroup$
    – Vagabond
    Jun 15, 2010 at 19:06
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    $\begingroup$ I think the prove is just squaring and computing the integral, and estimating the remaining terms by $\frac{|c_k| |c_l|}{\Delta |l -k|}$. Then one applies Cauchy-Schwarz twice to get the inequality. But I might have made a mistake on my scrap paper. $\endgroup$
    – Helge
    Jun 15, 2010 at 21:40
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    $\begingroup$ I strongly recommend that you modify the title of the question --- it's preferable if the title is a question, and at least it could have more info than "Salem Inequality". Something like: How do you prove (and what is another name for) the "Salem Inequality"? $\endgroup$ Jun 15, 2010 at 22:30
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    $\begingroup$ @Vagabond, Salem is famous by his contributions in the theory of Fourier series (www-history.mcs.st-andrews.ac.uk/Biographies/Salem.html), so the inequality is most probably a spacial one of them. Then it should be reflected in any comprehensive treatise on Fourier series (e.g., Zygmund's). $\endgroup$ Jun 15, 2010 at 23:43

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Following a cue from Wadim, this inequality is Theorem 9.1 in Chapter 5 of Zygmund's Trigonometric series, 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.)

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  • $\begingroup$ Thankyou for pointing out the exact reference. $\endgroup$
    – Vagabond
    Jun 16, 2010 at 10:16

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