Behavior of Solutions of Convolution Equation at Infinity

  • V. V. Volchkov

Abstract

Let ϕɛ′(ℝ n ), ϕ ≠ 0 and fL loc(ℝ n ) 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 fL(ℝ n ), 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 ℝ n . As \(\widehat f\) is continuous we obtain f = 0.

Keywords

Convolution 

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Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • V. V. Volchkov
    • 1
  1. 1.Department of MathematicsDonetsk National UniversityDonetskUkraine

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