The Fourier inversion and the Riemann functional equation

We prove that the Riemann functional equation can be recovered by the Mellin transforms of essentially all the absolutely integrable functions. The present analysis shows also that the Riemann functional equation is equivalent to the Fourier inversion formula. We introduce the notion of a λ-pair of absolutely integrable functions and show that the components of the λ-pair satisfy an identity involving convolution type products.