$\displaystyle \int_T \ t^n \phi(t) \, dt = \left( \int_T \ t \phi(t) \, dx \right)^n $
n is an integer .
what could $\displaystyle \phi(t) \$ be ?
Consider:
$\displaystyle \int_0^t x^n \phi(x) \, dx = \left( \int_0^t x\, \phi(x) \, dx \right)^n$ and differentiate both sides with respect to t:
$\displaystyle t^n \phi(t) = n \left( \int_0^t x\, \phi(x) \, dx \right)^{n-1} (t \, \phi(t) \, )$
$\displaystyle \Rightarrow t^{n-1} = n \left( \int_0^t x\, \phi(x) \, dx \right)^{n-1}, ~ \phi(t) \neq 0$
$\displaystyle \Rightarrow \frac{t\, \phi(t) }{n^{n-1}} = \int_0^t x\, \phi(x) \, dx$ .....
Well, since this contains two such careless mistakes, I though I'd better set the record straight.
The result in the last line should of course be $\displaystyle \frac{t}{n^{1/(n-1)}} = \int_0^t x\, \phi(x) \, dx$.
Then:
Let $\displaystyle k = \frac{1}{n^{1/(n-1)}} = n^{1/(1-n)}$ and differentiate both sides wrt t:
$\displaystyle k = t \phi(t) \Rightarrow \phi(t) = \frac{k}{t}$.
thanks , but there is something wrong , 'cause $\displaystyle \phi(t) $ should be independent of n . the equality is required to be true for any positive integer ... so - obviously - $\displaystyle \phi(t) $ should be independent of n .
here is what i did :
assume $\displaystyle \phi(t) $ is smooth and analytic at $\displaystyle t=t_0 $ , then it can be expanded in terms of a power series . now setting the correct relations on both RH and LH parts , and integrating over T , we end up with something like this :
$\displaystyle \sum^{\infty}_{r=0}\frac{b_r}{r+n+1}= \left(\sum^{\infty}_{r=0}\frac{b_r}{r+2}\right)^{n }$
$\displaystyle n=0,1,2,3 ...... $
now the program is to solve for $\displaystyle b_r $ in general .. so , is this doable ?
formally - if i'm not mistaken - the problem transforms to finding a set of measure spaces whose measure ($\displaystyle \ ds =\phi(t)dt $) and nth norm $\displaystyle
\left\|t\right\|_n = \left( \int \ t^n\ ds \right)^\frac{1}{n}
$ satisfy :
1- $\displaystyle \int ds =1$
2- $\displaystyle \left\|t\right\|_n$=$\displaystyle \left\|t\right\|_1$ ,$\displaystyle n=2,3,4 .... $
i think the problem went harder , but more formal .
so , i have been discussing the problem with the guys in physics forums , here is the link
a problem in Lp spaces .
i think it helps to look at it .