Use of Dirac delta to define an inverse

[lukluk]

I was wondering which are the properties of functions defined in such a way that

∫dx f(y-x) g(x-z) = δ(y-z)

where δ is Dirac delta and therefore g is a kind of inverse function of f (I see this integral
as the continuous limit of the product of a matrix by its inverse, in which case the δ becomes the identity matrix). Does anyone know where this type of "inverse functions" are
discussed? or how to obtain the function g given f ?

 

[lukluk]

update:

I found that taking the Fourier transform of both sides one can show that
g(y)=∫dk (1/f(k)) e^ikx
up to some factor of 2π, where f(k) is the Fourier transform of f(x).