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 Zh. Mat. Fiz. Anal. Geom., 2018, Volume 14, Number 3, Pages 336–361 (Mi jmag703)

The extended Leibniz rule and related equations in the space of rapidly decreasing functions

Hermann Königa, Vitali Milmanb

a Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
b School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel

Abstract: 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.

Key words and phrases: rapidly decreasing functions, extended Leibniz rule, Fourier transform.

 Funding Agency Grant Number MINERVA Foundation Alexander von Humboldt-Stiftung United States - Israel Binational Science Foundation (BSF) 200 6079 Israel Science Foundation 387/09 The first author is supported by Minerva. The second author is supported in part by the Alexander von Humboldt Foundation, by ISF grant 387/09 and by BSF grant 200 6079.

DOI: https://doi.org/10.15407/mag14.03.336

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MSC: 39B42, 47A62, 26A24
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Citation: Hermann König, Vitali Milman, “The extended Leibniz rule and related equations in the space of rapidly decreasing functions”, Zh. Mat. Fiz. Anal. Geom., 14:3 (2018), 336–361

Citation in format AMSBIB
\Bibitem{KonMil18} \by Hermann~K\"onig, Vitali~Milman \paper The extended Leibniz rule and related equations in the space of rapidly decreasing functions \jour Zh. Mat. Fiz. Anal. Geom. \yr 2018 \vol 14 \issue 3 \pages 336--361 \mathnet{http://mi.mathnet.ru/jmag703} \crossref{https://doi.org/10.15407/mag14.03.336} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000450683100005}