Deconvolution when the convolution kernel has no inverse

A short study on the general deconvolution problem when the kernel has no inverse proves that a priori information on the signal to be restored is a necessary condition for deconvolution. The proposed deconvolution method uses the following information: the signal to be restored has a bounded support; this support is known or is inside a known interval. This method concerns the convolution kernels whose Fourier transform has a cutoff frequency. This type of kernel has a wide practical field. The image restoration and the processing of "the principal value solution" of the deconvolution problem are the most characteristic elements. The method is derived from the general Liouville-Neuman theory of solving integral equations. This new method incorporates and extends Ville\´s analytic continuation and Van Cittert\´s successive convolution method. The iterative deconvolution algorithm is very simple. The advantages of this method are shown by numerical results and, in particular, by an experimental spectroscopic application.