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Structural theorems for quasiasymptotics of distributions at the origin. (English) Zbl 1189.46032

The quasiasymptotics of distributions at the origin is defined with respect to the function \(t^aL(t)\), where \(a\in\mathbb{R}\) and \(L\) is a function slowly varying at zero. It is easy to prove that if a tempered distribution \(T\) has quasiasymptotics at zero with respect to \(t^aL(t)\) in \(S'\), then \(T\) has it in \(D'\), as well. In the meantime, it was shown by some authors that, with some limitation on \(L\) and \(a\), the converse is also true.
In this paper, it is simply proved (Theorem 6.1): Let \(f\in S'\). If \(f\) has quasiasymptotics at \(0\) in \(D'\), then \(f\) has quasiasymptotics at \(0\) in \(S'\). The main tool for the proof is a new class of functions, i.e., the class of asymptotically homogeneous functions with respect to a slowly varying function which is elaborated in detail. With this result, an open problem is completely solved.

MSC:

46F05 Topological linear spaces of test functions, distributions and ultradistributions
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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