 The convolution of the compactly supported function "f" and the distribution "T" is then the function defined for each formula_108 by

It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function.If the distribution "T" has compact support then if "f" is a polynomial (resp.an exponential function, an analytic function, the restriction to formula_85 of an entire analytic function on formula_155, the restriction to formula_85 of an entire function of exponential type in formula_155) then the same is true of formula_113.