Stability in generalized functions (Q657131)

Summary: We consider the additive functional equation in \(n\) independent variables \[ f(\sum^n_{i=1} x_i) = \sum^n_{i=1} f(x_i) + \sum^n_{i=1} f(x_i - x_{i-1}) \] in the spaces of generalized functions. Making use of the heat kernels, we find general solutions and solve the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using mollifiers, we extend these results to the space of distributions.