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

摘要

We solve the extended Leibniz rule T(f g) = Tf Ag + Af Tg foroperators T and A in the space of rapidly decreasing functions in bothcases of complex and real-valued functions. We find that Tf may be a linearcombination of logarithmic derivatives of f and its complex conjugate f withsmooth coefficients up to some finite orders m and n respectively and Af =fm f n. In other cases Tf and Af may include separately the real and theimaginary part of f. In some way the equation yields a joint characterizationof the derivative and the Fourier transform of f. We discuss conditions whenT is the derivative and A is the identity. We also consider differentiablesolutions of related functional equations reminiscent of those for the sineand cosine functions.