Matching prescribed integral moments

Given integral functionals $\Phi_1,\ldots,\Phi_m$ on a class of functions, moment matching asks for $u$ such that

\Phi_j(u)=d_j,\qquad 1\le j\le m.

The moment map in this sense is $\Phi(u)=(\Phi_1(u),\ldots,\Phi_m(u))$ . The functionals may be linear weighted integrals or integrals of nonlinear expressions such as $u^2$ .

Function-valued targets

For profiles $u(r,\eta)$ , one may prescribe $\Phi_j(u)(\eta)=d_j(\eta)$ for every $\eta$ . There are finitely many rows, but each row is a function. Solving at finitely many sample values does not establish this identity. Corrections built from fixed radial bumps and parameter-dependent coefficients reduce the problem to a parameterized finite system. This integral moment map is distinct from a Hamiltonian moment map in symplectic geometry.