L^\infty $ Error Bounds in Partial Deconvolution of the Inverse Gaussian Pulse

When a $C^\infty $ approximation to the Dirac (5-function, in the form of an inverse Gaussian pulse, is used as input into a linear time invariant system, the output waveform is an approximation to that system's Green's function, in which the singularities have been smoothed out. The ill-posed deconvolution problem for the output signal aims at reconstructing these singularities. By exploiting the smoothing properties of the inverse Gaussian kernel, we prove that partial deconvolution of the output waveform, given $L^2 $ a priori bounds on the data noise and the unknown Green's function, results in $L^\infty $ error bounds for the regularized solution and its derivatives. Consequently, when the $L^2 $ norm of the output noise is sufficiently small, partial deconvolution is a pointwise reliable $C^\infty $ function, which in turn approximates the desired Green's function in many applications.