Suppose we have a sequence $(a_k)_k$ in $\mathbb{C}$. I am wondering under which conditions it is true that there is a function $f \in L^1 \left[ 0, 1 \right]$ such that $a_k$ is the k-th moment of $f$.
I'm aware of Hausdorff's solution to the moment problem which gives a condition on the sequence $(a_k)_k$ which ensures the existence of a Borel-measure whose k-th moment is $a_k$ for every $k \geq 0$ but I also need this measure to be continuous with respect to Lebesgue-measure.
The only thing which I was able to show is that such a sequence converges to $0$. This follows easily from Lebesgue's Theorem.
To provide some context: If $f(z) = \sum f_k z^k, g(z) = \sum g_k z^k$ are functions in the Bergman space, then their inner product is given by $$ \langle f, g \rangle = \int_\mathbb{D} f \overline{g} d \lambda^2 = \sum (k+1)^{-1} f_k \overline{g_k}$$ where $\mathbb{D}$ is the unit disk and $\lambda^2$ denotes the 2-dimensional Lebesgue-measure.
I was wondering under which conditions a weighted inner product, say $\sum \omega_k f_k \overline{g_k}$, admits a similar integral representation.
Nevertheless, at this point I am only interested in the first question which I think is interesting on it's own.