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I am studying for the quals in January and am working through the following problem:

Let $f:[0,1] \to [0,1]$ be a continuous function with the property that $\displaystyle\int_0^1 f(x)x^n \, \textrm{d}x = \dfrac{1}{n+2}$. Show that $f(x)\equiv x$.

I was thinking to do integration by parts, but this does not seem to yield anything. Any suggestions would be much appreciated!

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For what range of $n?$ – Igor Rivin Dec 20 '13 at 3:51
note that you don't have any assumptions about differentiability so, strictly speaking, integration by parts isn't available to you – crf Dec 20 '13 at 3:55
Igor, the question puts no stipulations on the value of $n$. – Tyler Clark Dec 20 '13 at 4:01
up vote 6 down vote accepted

Consider $g(x)=f(x) -x$, then $$ \int g(x)p(x)dx = 0 $$ for every polynomial $p$. Now by Weierstrass Approximation, and the continuity of the integral, it follows that $$ \int g^2(x)dx = 0 $$ Since $g$ is continuous, it follows that $g \equiv 0$.

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Thank you! My second thought was Weierstass, but I wasn't sure if there was a better route to go. – Tyler Clark Dec 20 '13 at 4:26
Why do we need that integral condition? continuity of $f$ is sufficient to have a fixed point. – Un Chien Andalou Dec 20 '13 at 5:52
Where does the first line come from? – Chris Dec 20 '13 at 11:34
@Tojamaru : Not sure what you mean, but the question does not ask for a fixed point, but for $f$ to be identically equal to $x$. – Prahlad Vaidyanathan Dec 20 '13 at 11:36
@Chris : The given condition says that $$\int f(x)x^ndx = \int x\cdot x^ndx$$ – Prahlad Vaidyanathan Dec 20 '13 at 11:36

Perhaps this helps... $$(n+2)\int_0^1f(x)x^ndx=1$$ Then $$\int_0^1(n+2)f(x)x^ndx=1$$ You can see the power rule embedded in there now, except $x^n$ would need to be $x^{n+1}$. So what must $f(x)$ be?

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I would say, (a) Prove it for $f$ polynomial. (b) use Stone-Weierstrass to approximate by polynomials, and then use (a) to show that the coefficients other than the linear term are arbitrarily small.

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