Chebyshev polynomials with non-negative constants Please let help me solve the following problem that I encountered while engaging in my research.
I'm dealing with a class of functions, in which each function has a unique series representation of the form
$$f(t) = \sum_{k = 0}^{\infty}a_kt^k$$ with each $a_k > 0$ and $\sum_{k = 0}^{\infty}a_k < \infty$.
Now, in order to utilize some algorithm I need to approximate any given function in the above class as the "conical" combination a "finite" basis. i.e.,
$$f(t) \approx \sum_{k = 0}^{N}b_kg_k(t)$$ with each $b_k > 0$.
I can easily set $g_k(t) = t^k$, but then the the series does not converge to $f(t)$ when $N$ is not very large.
My question is: Is there a better set of basis functions ($g_k(t)$ s) that I can use to approximate $f(t)$ more efficiently? To my understanding, this has something to do with approximating with the Taylor series vs approximating with Chebyshev polynomials.
 A: There's a decomposition of a power in to a Chebyshev basis polynomial (3.35).  ie: $$ x^N = 2^{1 - N} \sum_{k=0}^{N/2} \binom{N}{k} T_{n-2k}(x)$$
Where $T_m(x)$ is the $m$th Chebyshev polynomial of the first kind, and $\binom{a}{b}$ are binomial coefficients.  It should be straightforward to see how to convert a normal polynomial in to a Chebyshev basis using the above identity and a bit of algebra (although this isn't the fastest way to do it).
From the fact that $a_k > 0$ and $\binom{N}{k} > 0$ and $a_k * \binom{N}{k} > 0$, you can show that representing your polynomial in a Chebyshev basis should still preserve the positive nature of the coefficients.  (That is, your $b_k > 0$).
Polynomials represented in a Chebyshev basis have a variety of useful numerical properties.  Or put simply, they tend to be numerically stable in a way that monomial polynomials (that is, linear sums of powers of $x$) are not.
edit: worth mentioning, Chebyshev polynomials are defined on $[-1,1]$.  If you're interested in other domains, you'll need to scale and shift it to that domain first.
