Approximating a function with sums of powers

Question

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.

Answer 1

Orthogonalize your powers on your interval using the Gram-Schmidt algorithm, and then apply the usual Fourier formulas for expansions in an orthogonal system.

Answer 2

You may want to restrict $x$ to be positive and then expressions of the type you are dealing with called "posynomials" if the $c_k$ are also positive. Posynomials are convex and they are indeed used in optimization and are treated in Boyd and Vandenberges "Convex optimization", but there the exponents are fixed.

It may help to write a posynomial as a sum of "exponential of affine functions" in the variable $y = \log(x)$ $$ f(x) = \sum_k c_k x^{a_k} = \sum_k e^{a_k y + \log(c_k)}. $$ Now, your approximation task looks like a job for Prony's method for approximating function with exponential sums. Note that the usual Prony method uses even complex $a_k$ but indeed the values of the $a_k$ are free variables in this method.