Behavior of Solutions of Convolution Equation at Infinity

Abstract

Let ϕ ∈ ɛ′(ℝⁿ), ϕ ≠ 0 and f ∈ L _loc(ℝⁿ) be a nonzero function satisfying the equation

$$ \left( {f * \phi } \right)\left( x \right) = 0, x \in \mathbb{R}^n . $$

(3.1)

Then f cannot decrease rapidly on infinity. For instance, if f ∊ L(ℝⁿ), from (3.1), (1.6.2) we have $\widehat f \cdot \widehat \varphi = 0$. Since $\widehat \varphi$ is an entire function the set $\{ x \in {\mathbb{R}^n}:\widehat \varphi (x) = 0\}$ is dense nowhere in ℝⁿ. As $\widehat f$ is continuous we obtain f = 0.