My question is simple, that
The Fourier Legendre series make use of Legendre polynomial defined on [-1,1] http://mathworld.wolfram.com/Fourier-LegendreSeries.html
What if the variable x that the function consists of is outside of [-1,1] as [a,b]? Then one should anticipate that the corresponding Legendre polynomial be defined on [a,b], correct?
This brings to my question: how to find the Legendre polynomial over the interval [a,b]? I know that for the case of [0,1], the corresponding Legendre polynomial is called the shifted Legendre polynomial. But how to generalise the same idea to [a,b] in a direct, simple, yet computational efficient way? For instance, is it possible to perform a direct transformation using the Legendre polynomial defined on [-1,1] then convert it to [a,b]?
Thank you!
Polynomials are defined everywhere. But if what you want are orthogonal polynomials on the interval $[a,b]$ analogous to the Legendre polynomials which are orthogonal on $[-1,1]$, you may want to take $$f_n(x) = P_n\left(\frac{2x-a-b}{b-a}\right)$$ where $P_n$ are the Legendre polynomials.