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Let $\mathcal P$ be the collection of trigonometric polynomial with $L^p[0,1]$ norm one, $p>2$, then by changing the sign of coefficients of polynomials in $\mathcal P$ randomly, can we produce polynomials having arbitrary large $L^q[0,1]$ norm for $2<q\leq p?$ What is a good reference for this kind of material?. Thanks.

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    $\begingroup$ I'm not sure of the answer to your question off-hand (though I'd guess yes), but the place to start looking for this kind of material is Kahane's book "Some Random Series of Functions". $\endgroup$ Feb 19, 2014 at 11:47
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    $\begingroup$ Is $P$ itself random in some sense? If not, isn't the answer obviously "no" for $q = p$ because the signs could already have been chosen to maximize the norm? Or by "large" do you mean "not too small"? $\endgroup$
    – Noah Stein
    Feb 19, 2014 at 14:44
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    $\begingroup$ @NoahStein: The question is pretty vague as stated, and I assumed "not too small" was an allowable interpretation of "large" (and, for that matter, that "has" can mean "has on average", "has with not too small probability", etc.). $\endgroup$ Feb 19, 2014 at 14:59
  • $\begingroup$ I have edited the question. Thanks. $\endgroup$
    – user85038
    Feb 20, 2014 at 4:24
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    $\begingroup$ Rudin-Shapiro polynomials: coefficients $\pm 1$, all norms about $\sqrt n$. Dirichlet Kernels: all coefficients $1$, $\|...\|_{L^q}\approx n^{1-\frac 1q}$. $\endgroup$
    – fedja
    Mar 27, 2014 at 23:25

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The question is whether the trigonometric functions form an unconditional basis for $L^p$. Unfortunately, I have no access to the literature at the moment but if memory serves me well the answer to the basis problem is negative (and so positive for your problem) and can be found in "Classical Banach spaces" by Lindenstrauss and Tzafriri (probably vol. 2).

Edit: I have now located the result that the trigonometric functions are not an unconditional basis in any $L^p$ (with the obvious exception) in Heil:A basis theory primer.

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