Singular system of a compact operator

A singular system $({\sigma }_j,u_j,v_j)$ for a compact operator $A:X\to Y$ has positive singular values ${\sigma }_j\to 0$ and orthonormal vectors satisfying $A{v}_j={\sigma }_j{u}_j$ and $A^{\ast }{u}_j={\sigma }_j{v}_j$. It diagonalizes spectral regularization methods.

The displayed singular value system applies when the Fourier multipliers $c_n$ are nonzero. The phase in the left singular vector is necessary for complex kernels. If a multiplier vanishes, its mode instead belongs to the corresponding null spaces and is excluded from the positive singular system.