The deconvolution operation in convex analysis: An introduction

Performing the infimal convolution of two functions is a frequent and useful operation in Convex Analysis: it is, to great extent, the dual operation of the addition; it serves (like other {open_quotes}convolutions{close_quotes} in Analysis) to regularize functions; it has nice geometrical and economic interpretations. The deconvolution of a (convex) function by another one is a new operation, firstly defined in clear-cut manner, which is to the infimal convolution what the subtraction is to the addition for real numbers; it appears in conjugating the difference of convex functions; it serves in solving explicitly convolution equations; it has an interpretation in terms of subtraction of epigraphs. Since its introduction, the deconvolution operation has been studied more thoroughly by the author and his former students or associates. What we intend to present here is a short (and, hopefully, pedagogical) introduction to the deconvolution operation, in a simplified setting. This can be viewed as a complement to chapter IV and X in the book.