Consider a set of integrable functions on the interval $(0,1)$.

Let's introduce an operation $\operatorname{eval}f=\int_0^1 f(x)\,dx$ (which is the mean value of the function).

In such system the function $f(x)=x$ will play the role of an umbra with moments $1,1/2,1/3,1/4,...$ because $\int_0^1 x^n \, dx=\frac 1{n+1}$.

My question is: is it always possible to find a function $f(x)$ such that $\int_0^1 f(x)^n \, dx = a_n$ where $a_n$ is an arbitrary sequence, for instance, Bernoulli numbers? If so, what function would be a representation of Bernoulli umbra?