Behavior of Solutions of Convolution Equation at Infinity

V. V. Volchkov
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V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 3 in Integral Geometry and Convolution Equations, 2003, pp 191-200 from Springer

Abstract: Abstract Let ϕ ∈ ɛ′(ℝ n ), ϕ ≠ 0 and f ∈ L loc(ℝ n ) be a nonzero function satisfying the equation 3.1 $$ \left( {f * \phi } \right)\left( x \right) = 0, x \in \mathbb{R}^n . $$ Then f cannot decrease rapidly on infinity. For instance, if f ∊ L(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 (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0023-9_16

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DOI: 10.1007/978-94-010-0023-9_16

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