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!


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.