My question is about deriving constraints for coefficients of a Chebyshev series which represents a CDF.
Let $F(x)$ be the cumulative distribution function for $x\in [-1,1]$. Accordingly we know that:
$$ F(-1) = 0, \quad F(1) = 1, \quad 0 \le F(x) \le 1, \quad \frac{\partial}{\partial x}F(x) \ge 0$$
If we represent $F(x)$ as a Chebyshev series
$$F(x) = \sum_n a_n T_n(x)$$
Then we may use the fact that
$$T_n(1) = 1, \quad T_n(-1) = (-1)^{n}$$
to derive the constraints that
$$ F(1) = \sum_n a_n T_n(1) = \sum_n a_n = 1$$ $$ F(-1) = \sum_n a_n T_n(1) = \sum_n (-1)^{n} a_n = 0$$
My question is this:
Can we use the inequalities given for $F(x)$ to derive any additional constraints on the coefficients $a_n$?
Thanks for any help!
