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A Tauberian theorem for distributional point values. (English) Zbl 1169.46019

The definition of the value of a distribution at a point was first given by S.Łojasiewicz [Stud.Math.16, 1–36 (1957; Zbl 0086.09405)]. A distribution \(f\in {\mathcal D}'({\mathbb R})\) has the value \(\gamma\) at the point \(x_0\in\mathbb{R}\) if the limit \(\lim_{\varepsilon\to0}f(x_0+\varepsilon x)\) exists in \({\mathcal D}'({\mathbb R})\) and is equal to the constant \(\gamma\). This is denoted by \(f(x_0)=\gamma\).
In the present paper, the authors prove that if \(f\in {\mathcal D}'(a,b)\) is the distributional limit of an analytic function \(F\) defined in \((a,b)\times (0,R)\), and \(F(x_0+i y) \to \gamma\) as \(y \to 0^+\), and \(f\) is distributionally bounded at \(x=x_0\), then \(f\) has the value \(\gamma\) at the point \(x_0\).

MSC:

46F20 Distributions and ultradistributions as boundary values of analytic functions

Citations:

Zbl 0086.09405
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