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 )

### 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

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.