Inversion formulas for tridiagonal Toeplitz matrices are used to give exact inversion formulas for rank one convolution operators. These formulas are used to analyze the existence and behavior of the inverse in both the diagonally dominant and non-diagonally dominant cases.
David C. Wilson,
"On The Decomposition And Inversion Of Convolution Operators", Proc. SPIE 1098, Aerospace Pattern Recognition, (30 August 1989); doi: 10.1117/12.960429; https://doi.org/10.1117/12.960429