One can approximate an analytic $f: \mathbb R\to\mathbb R$ with Chebyshev polynomials $T_n$ or with Taylor polynomials. In applications one usually prefers Chebyshev ones because they would converge faster on a given interval. For some applications (including the one I'm dealing with right now) one can use a somewhat a bit broader set of functions, such as "sums of powers": $$\Sigma_{k=0}^n c_k x^{\alpha_k}$$ where $\alpha_k$ are not necessarily integers.

I wonder where I could look for a convergence study of expressions like that. Specifically, I'd like to know how for a given $n$ to compute a sum of powers that approximates $f$ best on a given interval (in any norm $\|\cdot\|_1$ or $\|\cdot\|_2$).

EDIT: I probably should clarify the question. The set of powers $\alpha_k$ is not fixed prior to approximation. The goal is not only to pick the best coefficients $c_k$ but also the powers $\alpha_k$ for the given $f$ so that the $n$-term approximation of $f$ would have the least error.

Here's an illustration. Suppose $f=\sqrt x, x\in[1,3]$. One can certainly approximate $f$ with a Taylor polynomial centered at $2$ or with a Chabyshev polynomial, etc. However, the best approximation in the above form would be, well, $\sqrt x$: $c_0=1, \alpha_0=\frac{1}{2}$.

Given that uncountable choice of $\alpha$s it's not clear how one can orthogonalize anything. The goal seems to be, for the given $f$ and $n$, to identify the sequence $\alpha_0,...,\alpha_n$ which would span a plane closest to $f$, and then compute the coefficients that would produce the best approximation.