Suppose that $f$ is a many-times integrable function on $[-1, 1]$. We can consider integral moments of $f$, given by $$ I_n(f) := \int_{-1}^1 \big( f(x) \big)^n dx.$$ My question is: to what extent do the moments $\{I_n(f)\}_{n \in \mathbb{N}}$ determine $f(x)$?

Clearly this is hard to answer if some conditions are not imposed on $f$. For instance, changing $f$ at any single value doesn't affect any of the moments. So a better question is to ask to what extend do the moments determine a *nice* function $f$ for suitable choices of *niceness*.

In particular, does the sequence of moments $\{ I_n(f)\}_{n \in \mathbb{N}}$ determine $f$ completely if $f$ is continuous and positive? Or perhaps if $f$ is smooth and positive? Or perhaps if $f$ is analytic and positive?