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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.
Received: 08.02.2018
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
Linking options:
https://www.mathnet.ru/eng/jmag703 https://www.mathnet.ru/eng/jmag/v14/i3/p336
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Abstract page: | 229 | Full-text PDF : | 59 | References: | 33 |
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