The finite Legendre transformation of generalized functions (Q1293539)

Starting with the conventional finite Legendre transformation \[ l\{f(x)\} = F(n) =\int_{-1}^1 P_n(x)f(x) dx,\quad n=0,1,2,\dots, \] where \(P_n\) are the well-known Legendre polynomials, the authors introduce the generalized finite Legendre transformation on the space \(\mathcal L'(-1,1)\) as \[ (l'f)(n) = F(n) = \langle f,P_n\rangle,\quad n=0,1,2,\dots. \] Here \(\mathcal L'(-1,1)\) is the dual space of the space \(\mathcal L(-1,1)\) of all complex-valued infinitely differentiable functions \(\phi\) defined on the open interval \((-1,1)\) such that \[ \gamma_k(\phi)=\sup_{-1<x<1}|R_x^k\phi(x)|<\infty, \] for every nonnegative integer \(k\), \[ R_x = D(x^2 -1)D,\quad D=\tfrac {d}{dx}. \] The two main results of the paper are the uniqueness theorem for the generalized finite Legendre transform and its inversion given as \[ \lim_{N\to \infty}\sum_{m=0}^N (m+\tfrac 12)F(m)P_m(x) = f(x) \] in the sense of the convergence in \(\mathcal L'(-1,1)\). Finally, the authors derive an operation transform formula for the generalized finite Legendre transformation and the differential operator \(R_x^k\), \(k=0,1,\dots\) and demonstrate its application for solving a boundary-value problem written in a spherical coordinate system.