Stability of Trigonometric Functional Equations in Generalized Functions

Ulam Problem for the Cosine Addition Formula in Sato Hyperfunctions

“…Note that inequality (6) means that inequality (3) holds in almost everywhere sense. In [22][23][24][25][26], some stability problems of several functional equations including inequality (6) were considered in various spaces of generalized functions including Schwartz distributions. In [24][25][26], for example, replacing and by distributions and V in inequality (6) we have considered…”

“…where ( , ) = + , , ∈ R , ∘ and ⊗ denote the pullback and tensor product of generalized functions. Inequality 7cannot be considered as a complete formulation in the sense of generalized functions because the differences are assumed to be classical bounded measurable functions and all the previous results in [24][25][26] have the same formulations as in (7). Due to Schwartz [19] the space ∞ (R ) of bounded measurable functions was generalized to the space D ∞ (R ) of bounded distributions.…”

On the Stability of Trigonometric Functional Equations in Distributions and Hyperfunctions

“…Unfortunately, there was no use of these results until 1978 when Rassias [7] treated with the inequality of Aoki [3]. Following Rassias' result, a great number of papers on the subject have been published concerning numerous functional equations in various directions [6][7][8][9][10][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. In 1990 Székelyhidi [24] has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with stability questions for the sine and cosine functional equations.…”