Stability of a Quadratic Functional Equation in the Spaces of Generalized Functions

Abstract

Making use of the pullbacks, we reformulate the following quadratic functional equation: f(x+y+z)+f(x)+f(y)+f(z)=f(x+y)+f(y+z)+f(z+x) in the spaces of generalized functions. Also, using the fundamental solution of the heat equation, we obtain the general solution and prove the Hyers-Ulam stability of this equation in the spaces of generalized functions such as tempered distributions and Fourier hyperfunctions.