Link between the coefficients of series expansion and an expansion based on a sub-interval?

Question

Consider a finite series expansion for a continuous function $f(x)$. For example using Legendre Polynomials, \begin{equation} f(x) = \sum_{i=0}^N f_n P_n(x) \quad x \in [-1,1] \end{equation} Is it possible to directly obtain the coefficients $g_m$ for a series expansion based on a sub-interval $[a,b]$, \begin{equation} g(x') = \sum_{i=0}^M g_m P_m(x') \quad x' \in [-1,1] \end{equation} where $x = a+(b-a)x'$ and g(x) = f(x) for $x \in [a,b] $, using knowledge of $f_n$? i.e. is there a simple relationship between $f_n$ and $g_m$?

Answer 1

Not a proof but if you simply substitute a general transformation of $x = \alpha + \beta x'$ into the original series expansion you can determine that there is a pattern which gives the coefficient transformation matrix $\bf{C}$ with, \begin{equation} C_{ij} = \begin{cases} \begin{pmatrix} j-1\\i-1 \end{pmatrix} \alpha^{j-i}\beta^{i-1} \quad j\geq i \\ 0 \quad \text{otherwise} \end{cases} \end{equation} where parenthesis indicates the binomial coefficient. Giving \begin{equation} \bf{g} = \bf{C} \bf{f} \end{equation} where $\bf{g}$ and $\bf{f}$ are vectors of the series coefficients.