In this lecture: https://ee227c.github.io/notes/ee227c-lecture6.pdf the author states that he wants to create a polynomial $P_{k}(a)$ in terms of Chebyshev polynomials of the first kind with the properties:
a) the Chebyshev polynomial, $T_k$, has to be rescaled to be in the range $[\alpha, \beta]$
b) at the origin, the desired $P_{k}(a)$ has to be $1$.
So he does a rescaling of the Chebyshev polynomial and states that $$ P_{k}(a) = \frac{T_{k}\left(\frac{\alpha + \beta - 2a}{\beta - \alpha}\right)}{T_{k}\left(\frac{\alpha + \beta}{\beta - \alpha}\right)} $$
I don't understand how he derives this result.
Thanks.
The function $f(a) = \frac{\alpha + \beta - 2a}{\beta - \alpha}$ of the affine transformation $P_k(a) = c T_k(f(a))$ maps $[\alpha,\beta]$ to $[-1, 1]$ (in an unsual way because $f(\alpha)=1, f(\beta)=-1$, normally one would use $\alpha - \beta$ in the denominator). The requirement $P_(0)=1$ is equivalent to $1 = cT_k(f(0))$ or $$c = \frac{1}{T_k(f(0))} = \frac{1}{T_{k}\left(\frac{\alpha + \beta}{\beta - \alpha}\right)}$$