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a) I am trying to find the decreasing rearrangement (DR) $f^*$ of the following function: $$f(x)=\sin(2x)+\sin(x)+2$$ in the interval $[0,2\pi]$. Admittedly, the $+2$ serves to get rid of negatives.

I learnt from this video to derive that if $f(x)=\sin(x)$ then $f^*(x)=\cos(x/2).$ This was pretty much the only source I found (and could comprehend) so far. Thank you for suggestions.

b) In general, I would like to be able to find the DR of (pretty much) any sum of sinusoidal functions, such that $$ f(x)=\sum_{k=0}^n \alpha_k \sin(2kx)+\sin(x)+n+1 $$ of which the above in (a) is the case for $n=1.$

I'm stuck, please help. Thanks in advance.

Jan Suchanek
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  • What does the DR of function $f$ refer to? – coffeemath Dec 23 '17 at 06:31
  • Well, Decreasing Rearrangement; definition see here: https://math.stackexchange.com/questions/731216/distribution-function-and-decreasing-rearrangement – Jan Suchanek Dec 23 '17 at 06:43
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    The definition itself (not just a link) belongs into your question, not into some comment. –  Dec 23 '17 at 07:02
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    Thanks; let $(X,dx)$ be a measure space and $f\in L^{p}(X,\mathbb{C})$; define its distribution function $F(\alpha)=meas({x\in X||f(x)|>\alpha})$ and the decreasing rearrangement $\alpha k=inf{\alpha>0|F(\alpha)<2k}$. Put in simpler (and probably less precise) words: "sort" all the values of $f$ decreasingly, thus getting a function "decreasing rearrangement" $f^*$ which has the same (local) maximum and minimum and integral in the interval in question. - Sorry for lack of lingo, not a professional at work. – Jan Suchanek Dec 23 '17 at 10:41
  • The intent was that you add this definition to the question, not in another comment. – Lutz Lehmann Dec 23 '17 at 12:16
  • The definition depends on a finite measure on $\Bbb R$. What is this finite measure in your case? – Lutz Lehmann Dec 23 '17 at 12:17
  • Still nobody interested? – Jan Suchanek Jan 22 '18 at 13:33

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