Approximation of functions with zero integrals over balls by linear combinations of the Helmholtz equation solutions (Q2901617)

The paper deals with questions of approximation of solutions of the convolution equation \(f*T=0\), \(T\) is radial, by linear combinations of solutions of equations \((\Delta+\lambda^2)^\eta u=0\), \(\eta\in\mathbb N\), such that \(u*T=0\), \(\Delta\)-the Laplace operator. It is proved that for an arbitrary open domain \(U\subset\mathbb R^n\) a function \(f\in C^\infty(U)\) with zero integrals over balls of radius \(r\) can be approximated in the \(C^\infty\)-topology by linear combinations of solutions of equations \((\Delta+\lambda^2)u=0\), which have zero integrals over balls of radius \(r\), \(\Delta\)-the Laplace operator. Analogues for two-point homogeneous spaces are considered.