In mathematics, the Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials P n α , β ( x ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} as kernels of the transform . The Jacobi transform of a function F ( x ) {\displaystyle F(x)} is
J { F ( x ) } = f α , β ( n ) = ∫ − 1 1 ( 1 − x ) α ( 1 + x ) β P n α , β ( x ) F ( x ) d x {\displaystyle J\{F(x)\}=f^{\alpha ,\beta }(n)=\int _{-1}^{1}(1-x)^{\alpha }\ (1+x)^{\beta }\ P_{n}^{\alpha ,\beta }(x)\ F(x)\ dx}
The inverse Jacobi transform is given by
J − 1 { f α , β ( n ) } = F ( x ) = ∑ n = 0 ∞ 1 δ n f α , β ( n ) P n α , β ( x ) , where δ n = 2 α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) n ! ( α + β + 2 n + 1 ) Γ ( n + α + β + 1 ) {\displaystyle J^{-1}\{f^{\alpha ,\beta }(n)\}=F(x)=\sum _{n=0}^{\infty }{\frac {1}{\delta _{n}}}f^{\alpha ,\beta }(n)P_{n}^{\alpha ,\beta }(x),\quad {\text{where}}\quad \delta _{n}={\frac {2^{\alpha +\beta +1}\Gamma (n+\alpha +1)\Gamma (n+\beta +1)}{n!(\alpha +\beta +2n+1)\Gamma (n+\alpha +\beta +1)}}}
Some Jacobi transform pairs
References