Journal of Mathematical Physics, Analysis, Geometry
2018, vol. 14, No 3, pp. 336-361   https://doi.org/10.15407/mag14.03.336     ( to contents , go back )
https://doi.org/10.15407/mag14.03.336

The Extended Leibniz Rule and Related Equations in the Space of Rapidly Decreasing Functions

Hermann König

Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
E-mail: hkoenig@math.uni-kiel.de

Vitali Milman

School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
E-mail: milman@post.tau.ac.il

Received February 8, 2018.

Dedicated to the 95th birthday of the great mathematician Vladimir Marchenko and to the 80th birthday of our friend and great mathematical physicist Leonid Pastur

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.

Mathematics Subject Classification 2000: 39B42, 47A62, 26A24.
Key words: rapidly decreasing functions, extended Leibniz rule, Fourier transform.

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