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  "body": "\n## Definition\n\nFix real numbers $\\alpha \u003e -1$ and $\\beta \u003e -1$, and let $w(x) = (1 - x)^\\alpha (1 + x)^\\beta$ for $-1 \u003c x \u003c 1$.\nLet $P_n^{(\\alpha, \\beta)}$ be the [[Jacobi polynomial|TRD7DSF2KR]] of degree $n$.\nLet $F \\colon (-1, 1) \\to \\C$ be a function such that $w F$ is integrable on $(-1, 1)$.\nThe *Jacobi transform* of $F$ is the sequence\n$$f(n) = \\int_{-1}^{1} w(x)\\, P_n^{(\\alpha, \\beta)}(x)\\, F(x)\\, dx, \\qquad n = 0, 1, 2, \\dots$$\n\n## Remarks\n\nLet $h_n = \\int_{-1}^{1} w(x)\\, P_n^{(\\alpha, \\beta)}(x)^2\\, dx \u003e 0$.\nFor $F$ 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 $L^2(w)$.\n\nLet $R[F] = w^{-1} \\frac{d}{dx}\\left( (1 - x)^{\\alpha + 1} (1 + x)^{\\beta + 1} F' \\right)$.\nIf $F$ 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 $-n(n + \\alpha + \\beta + 1) f(n)$.\n\nExample: $F = P_m^{(\\alpha, \\beta)}$ has $f(n) = 0$ for $n \\ne m$ and $f(m) = h_m$, by orthogonality.\n\nNon-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 $\\alpha, \\beta \u003e -1$ is required.\n\nFor $\\alpha = \\beta = \\lambda - 1/2$ one gets, up to constant factors, the [[Gegenbauer transform|TMNNRRRGAS]].\n",
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