Equations Involving the Mean of Almost Periodic Measures

Abstract

We use the theory of Fourier-Bohr series for almost periodic measures to looking for a complex-valued function f which is almost periodic on ℝ and satisfies equation

$$ f(x) = M_y [f(x - y)\mu (y)] + \nu * h (x), x \in \mathbb{R}. $$

(E)

In this context h is an almost periodic function on ℝ, μ is a positive almost periodic measure on ℝ and υ is a bounded measure also on ℝ. With a suitable choice of the measures μ and υ equation (E) becomes

$$ f(x) = \mathop {\lim }\limits_{t \to \infty } \frac{1} {{2t}}\int_{ - t}^t {f(x - y)g(y)} dy + \int_{ - \infty }^{ + \infty } {\varphi (y)h(x - y) dy} , x \in \mathbb{R}, $$

where g is an almost periodic function on ℝ and ϕ belongs to L ¹(ℝ).