Debnath and Harrell 1976 introduced the associated Legendre transform of a function $F(x)$ and its inverse, \begin{align} f(n,m) &= T_n^m[F(x)] = \int_{-1}^1 (1-x^2)^{-m/2} P_n^m(x) F(x) dx \\ F(x) &= (T_n^m)^{-1}[f(n,m)] = \sum_{n=0}^{\infty} \frac{2n+1}{2} \frac{(n-m)!}{(n+m)!} f(n,m) (1-x^2)^{m/2} P_n^m(x) \end{align} where $P_n^m(x)$ are the associated Legendre functions. This generalized the Legendre transform introduced by Churchill 1954, \begin{align} f(n) &= T_n[F(x)] = \int_{-1}^1 F(x) P_n(x) dx \\ F(x) &= T^{-1}[f(n)] = \sum_{n=0}^{\infty} \left(n+\frac{1}{2}\right) f(n) P_n(x) \end{align} The Churchill 1954 paper presents a powerful theorem (Theorem 3) on the product of Legendre transforms, $$ T^{-1}[f(n) g(n)] = H(x) $$ where $$ H(\cos{\theta}) = \frac{1}{\pi} \int_0^{\pi} F(\cos{\theta'}) \sin{\theta'} \int_0^{\pi} G(\cos{\lambda}) d\phi d\theta' $$ and $$ \cos{\lambda} = \cos{\theta} \cos{\theta'} + \sin{\theta} \sin{\theta'} \cos{\phi} $$ In the concluding remarks of the Debnath and Harrell 1976 paper, they say that work on a similar theorem for the associated Legendre transform is "under way" and will be reported in a subsequent paper. However, I have been unable to find a paper in which a similar theorem is presented.

Does anyone know if a similar theorem on the product of associated Legendre transforms exists in the literature? Please provide a reference if so.

Churchill, R. V. "The operational calculus of Legendre transforms." Studies in Applied Mathematics 33.1–4 (1954): 165–178.

Debnath, L. and Harrell, C. 1976. The operational calculus of associated Legendre transforms-I. Indian J. Pure and Appl. Math., 7(3): 278–291.


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