The Wigner transform of an integral kernel on the full line generalizes the Fourier transform of a translation kernel. The eigenvalue spectra of Hermitian kernels are related to the topographic features of their Wigner transforms. Two kernels whose Wigner transforms are equivalent under the unimodular affine group have the same spectrum of eigenvalues and have eigenfunctions related by an explicit linear transformation. Any kernel whose Wigner transform has concentric ellipses as contour lines, yields an eigenvalue problem which may be solved exactly.
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Knight, B. W., Jr. and Sirovich, L. (1982) The Wigner transform and some exact properties of linear operators. SIAM J. Appl. Math. 42, 378-389.