dependency on length of interval in Chebyshev coefficients

Question

Consider a function $g : [-1,1] \to \mathbb{R}$ and $c_n$ denotes the $nth$ Chebyshev coefficient of the function $g$. Moreover, $$c_n = \frac{2}{\pi}\int_{-1}^{1} T_n(x) g(x) \frac{1}{\sqrt{1-x^2}} dx = O(n^{-m}),\quad n\ge 1,$$ for some natural number $m$. Define $f$ as the restriction of $g$ on closed interval $[a,b] \subseteq [-1,1]$ parametrized over $[-1,1]$, that is $$ f(t) := g\left( \frac{b-a}{2}(t+1)+a \right),$$ and define $c_n'$ as the $n$th Chebyshev coefficient of $f$. Clearly the chebyshev coefficients $c_n'$ will have a dependency on the length of the patch $h=b-a$ (smaller $h$ implies faster decay of coefficients), however I am unable to figure out this dependency. Since $f$ and $g$ have the same regularity, we must have $c_n' = O(h^k n^{-m})$ for some unknown $k$. Any hints on how to solve for $k$?