The stability of the cosine equation

Source abstract

If δ > 0 \delta > 0 , G is an abelian group and f is a complex-valued function defined on G such that | f ( x + y ) + f ( x − y ) − 2 f ( x ) f ( y ) | ⩽ δ |f(x + y) + f(x - y) - 2f(x)f(y)| \leqslant \delta for all x , y ∈ G x,y \in G , then either | f ( x ) | ⩽ ( 1 + 1 + 2 δ ) / 2 |f(x)| \leqslant (1 + \sqrt {1 + 2\delta } )/2 for all x ∈ G x \in G or f ( x + y ) + f ( x − y ) = 2 f ( x ) f ( y ) f(x + y) + f(x - y) = 2f(x)f(y) for all x , y ∈ G x,y \in G .