Jacobi transform

Remarks

Let $h_n = \int_{-1}^{1} w(x)\, P_n^{(\alpha, \beta)}(x)^2\, dx > 0$. For $F$ in the weighted space $L^2(w)$, the inverse transform is the expansion $F = \sum_{n \ge 0} \frac{f(n)}{h_n} P_n^{(\alpha, \beta)}$, which converges in $L^2(w)$.

Let $R[F] = w^{-1} \frac{d}{dx}\left( (1 - x)^{\alpha + 1} (1 + x)^{\beta + 1} F' \right)$. If $F$ is smooth enough and the boundary terms vanish, two integrations by parts and the Jacobi differential equation give that the transform of $R[F]$ is $-n(n + \alpha + \beta + 1) f(n)$.

Example: $F = P_m^{(\alpha, \beta)}$ has $f(n) = 0$ for $n \ne m$ and $f(m) = h_m$, by orthogonality.

Non-example: for $\alpha = -1$ the weight $(1 - x)^{-1}(1 + x)^\beta$ is not integrable near $x = 1$, so already $F = 1$ has no transform; this is why $\alpha, \beta > -1$ is required.

For $\alpha = \beta = \lambda - 1/2$ one gets, up to constant factors, the Gegenbauer transform.