On a characterization of three functions by means of a generalized convolution (Q1078794)

Let \(f_ 0,f_ 1,f_ 2\) be three real Laplace originals, vanishing on the negative halfline. The generalized convolution \(f_ 0,f_ 1,f_ 2\) is defined by the formula \[ g(x_ 1,x_ 2)=\int^{\min \{x_ 1,x_ 2\}}_{0}f_ 1(x_ 1-t)f_ 2(x_ 2-t)f_ 0(t)dt,\quad x_ 1\geq 0,\quad x_ 2\geq 0. \] Let \(F_ 0,F_ 1,F_ 2,G\) be the Laplace transform of \(f_ 0,f_ 1,f_ 2,g\) correspondingly. The author obtains the explicit formulas for \(F_ 0,F_ 1,F_ 2\) in terms of G.