Stability in Generalized Functions

We consider the following additive functional equation with -independent variables: in the spaces of generalized functions. Making use of the heat kernels, we solve the general solutions and the stability problems of the above equation in the spaces of tempered distributions and Fourier hyperfunctions. Moreover, using the mollifiers, we extend these results to the space of distributions.

Generalized Fourier expansion in kernels of convolution operators on Fourier hyperfunctions

We prove that the kernels of surjective convolution operators on Fourier hyperfunctions (and on Fourier ultra-hyperfunctions) admit a basis of exponential solutions. The corresponding coefficient spaces are explicitly determined.

Asymptotic hyperfunctions, tempered hyperfunctions, and asymptotic expansions

We introduce new subclasses of Fourier hyperfunctions of mixed type, satisfying polynomial growth conditions at infinity, and develop their sheaf and duality theory. We use Fourier transformation and duality to examine relations of theseasymptoticandtemperedhyperfunctions to known classes of test functions and distributions, especially the Gel'fand-Shilov spaces. Further it is shown that the asymptotic hyperfunctions, which decay faster than any negative power, are precisely the class that allows asymptotic expansions at infinity. These asymptotic expansions are carried over to the higher-dimensional case by applying theRadon transformationfor hyperfunctions.