L(Infinity) Error Bounds in Partial Deconvolution of the Inverse Gaussian Pulse

摘要

When a Cat infinity approximation to the Dirac Delta-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 squared a priori bounds on the data noise and the unknown Green's function, results in L at infinity error bounds for the regularized solution and its derivatives. Consequently, when the L squared norm of the output noise is sufficiently small, partial deconvolution is a pointwise reliable Cat infinity function, which in turn approximates the desired Green's function in many applications. Keywords: pulse probing; impulse response; time domain deconvolution; ultrasonic flaw detection; optical fiber characterization; regularization; error bounds; partial deconvolution; Reprints.