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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!

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