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.
Given $R>0$, how well can we approximate $f$ as a linear combination of functions of the form $x^r$, where $r$ lies in $\lbrack 0,R\rbrack$? If $f$ is analytic, can we express $f$ as an integral $\int_0^R x^r d\mu(r)$?
As noticed in the previous remark, it is not true that every $f$ can be represented by a power series, $\sum c_kx^k$ because the sum of this series must be analytic. Same applies to the intergal $\int x^r d\mu(r)$. This function is analytic on intervals $I$ that do not contain zero.
However, there is an analog of Weierstrass theorem for non-integer powers $x^{r_k}, r_k>0$. This is called the Muntz-Sasz theorem. It says roughly speaking that the span of these monomials is dense in C[a,b], 0 < a < b, if and only if the series $\sum 1/r_k$ diverges.
Let us assume that your $f$ is continuous and compactly supported on the real line. Now for $\epsilon >0$, just consider the following convolution by a Gaussian mollifier $$ f_\epsilon(x)=\int e^{-\pi\epsilon^{-2}(x-y)^2} f(y) \epsilon^{-1}dy $$ Obviously the fonction $f_\epsilon$ is entire and also is converging uniformly towards $f$: we have $$ f_\epsilon(x)-f(x)=\int e^{-\pi y^2}\bigl( f(x+\epsilon y)-f(x)\bigr) dy, $$ so that $ \vert f_\epsilon(x)-f(x)\vert\le \int_{\vert y\vert \le \lambda} e^{-\pi y^2}\bigl\vert f(x+\epsilon y)-f(x)\bigr\vert dy + \int_{\vert y\vert \ge \lambda} e^{-\pi y^2}\bigl\vert f(x+\epsilon y)-f(x)\bigr\vert dy $ and $$ \vert f_\epsilon(x)-f(x)\vert\le \sup_{\vert x_1-x_2\vert\le \epsilon \lambda}\vert f(x_1)-f(x_2)\vert + \int_{\vert y\vert \ge \lambda} e^{-\pi y^2} dy 2\Vert f\Vert_{L^\infty(\mathbb R)}. $$ Choosing for instance $\lambda=\epsilon^{-1/2}$, and using uniform continuity of $f$, we prove uniform convergence. Note that it is an easy way to prove Stone-Weierstrass theorem on $\mathbb R$ since the entire function $f_\epsilon$ can be uniformly approximated by polynomials on compact sets.
