The Extended Leibniz Rule and Related Equations in the Space of Rapidly Decreasing FunctionsHermann König Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany Vitali Milman School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978,
Israel 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. |