Say I have a continuous function $f$ defined on a compact interval $I$ on the real line. As is well-known, I could approximate $f$ arbitrarily well by polynomials, i.e., we can express $f$ as an infinite series $\sum c_n x^n$.
Given $R>0$, how well can we express approximate $f$ as an integral a linear combination of functions of the form $\int_0^R x^r d\mu(r)$x^r$, where $r$ lies in $\lbrack 0,R\rbrack$? If not$f$ is analytic, how well can we approximate express $f$ by as an integral of that form?
(Note I put no requirements on what $\int_0^R x^r d\mu(r)$has to be for $x\notin I$ -- otherwise the answer to the first question would certainly be "no" in general.)?