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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

Entire Function Compact Support Precise Condition Uniqueness Theorem Analogous Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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