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The extended Leibniz rule and related equations in the space of rapidly decreasing functions
Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 3, pp. 336-361 Cet article a éte moissonné depuis la source Math-Net.Ru

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We solve the extended Leibniz rule $T(f\cdot g)=Tf \cdot Ag+Af\cdot Tg$ for operators $T$ and $A$ in the space of rapidly decreasing functions in both cases of complex and real-valued functions. We find that $Tf$ may be a linear combination of logarithmic derivatives of $f$ and its complex conjugate $\overline{f}$ with smooth coefficients up to some finite orders $m$ and $n$ respectively and $Af=f^{m}\cdot \overline{f}$ $^{n} $. In other cases $Tf$ and $Af$ may include separately the real and the imaginary part of $f$. In some way the equation yields a joint characterization of the derivative and the Fourier transform of $f$. We discuss conditions when $T$ is the derivative and $A$ is the identity. We also consider differentiable solutions of related functional equations reminiscent of those for the sine and cosine functions. 
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     title = {The extended {Leibniz} rule and related equations in the space of rapidly decreasing functions},
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Hermann König; Vitali Milman. The extended Leibniz rule and related equations in the space of rapidly decreasing functions. Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 14 (2018) no. 3, pp. 336-361. http://geodesic.mathdoc.fr/item/JMAG_2018_14_3_a4/

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