For a sequence of real numbers $s = (s_i)_{i \in n}$ let $M_s$ be the collection of functions $f:[0,1] \to [0,1]$ such that

$$(\forall i \leq n) \int_0^1 x^i f(x) dx = s_i$$

In other words, $M_s$ is the collection of density functions whose first $n$ moments are $(s_i)_{i \leq n}$.

I am interested in how much information knowing the first $n$ moments tells us about the function. Specifically, if we have the first $n$ moments of a function $f$ and we wish to approximate $f$ by $g$ with the same first $n$ moments, how badly could we be wrong?

To make this precise, for $p \in \{1, 2, \infty\}$ let $\alpha_p$ be the function of $s$

$$\alpha_p(s) = \sup_{f, g \in M_s} ||f - g||_p$$

So $\alpha(s)$ is an upper bound for how far off we could be if we approximate a function with first $n$ moments $s$ by another function with the first $n$ moments $s$.

Are there known lower bounds for the functions $n \to \sup_{|s| = n} \alpha_p(s)$ or $n \to \inf_{|s| = n} \alpha_p(s)$?

I feel this must have been studied so references are also welcome.

veryfar from an expert in this area I wasn't able to find an appropriate result after several hours of looking. Although I am hoping it would be easy/immediate to an expert. $\endgroup$ – Nate Ackerman May 24 '15 at 9:36