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Commutant and uniqueness of solutions of Duhamel equations. (English) Zbl 1475.46045

Summary: The Duhamel product for two suitable functions \(f\) and \(g\) is defined as follows: \[ (f\circledast g)(x)=\frac{\text{d}}{\text{d}x}\int\limits_0^xf(x-t)g(t) \,\text{d}t. \] We consider the integration operator \(J\), \(Jf(x)=\int\limits_0^xf(t)\text{d}t\), on the Fréchet space \(C^{\infty}\) of all infinitely differentiable functions in \([0,1]\) and describe in terms of Duhamel operators its commutant. Also, we consider the Duhamel equation \(\varphi\circledast f=g\) and prove that it has a unique solution if and only if \(\varphi(0)\neq 0\).

MSC:

46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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