On a direct method for proving the Hyers–Ulam stability of functional equations

We study the stability of an equation in a single variable of the formf(x)=af(h(x))+bf(−h(x))f(x)=af(h(x))+bf(−h(x)) with some conditions imposed on constants a, b and function h. The results are later applied for proving the stability (by use of a direct method) of a couple of functional equations in several variables. They let us not only shorten various approaches obtained during the last years but also improve the approximating constants.