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

Hermann König, Vitali Milman

Research output: Contribution to journalArticlepeer-review

Abstract

We solve the extended Leibniz rule T (f · g) = T f · Ag + Af · T g for operators T and A in the space of rapidly decreasing functions in both cases of complex and real-valued functions. We find that T f may be a linear combination of logarithmic derivatives of f and its complex conjugate f with smooth coefficients up to some finite orders m and n respectively and Af = f m · f n . In other cases T f 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.

Original languageEnglish
Pages (from-to)336-361
Number of pages26
JournalJournal of Mathematical Physics, Analysis, Geometry
Volume14
Issue number3
DOIs
StatePublished - 2018

Funding

FundersFunder number
Alexander von Humboldt-Stiftung
United States-Israel Binational Science Foundation200 6079
Israel Science Foundation387/09

    Keywords

    • Extended Leibniz rule
    • Fourier transform
    • Rapidly decreasing functions

    Fingerprint

    Dive into the research topics of 'The extended leibniz rule and related equations in the space of rapidly decreasing functions'. Together they form a unique fingerprint.

    Cite this