A characterization of product preserving maps with applications to a characterization of the Fourier transform

The chain rule as a functional equation

“…Note that (4) is not satisfied in example (3) and that (4.1) does not hold in example (2). The map T : C 1 (R) → C(R), Tf := |f | also fulfills (1) but does not satisfy (4.2).…”

“…anti-monotonicity or multiplicativity. The Fourier transform was studied in [1,2], geometry duality and the Legendre transform in [3,4], mixed volumes in [7]. This paper is concerned with characterizations of the derivative by the chain rule.…”

Characterizing the derivative and the entropy function by the Leibniz rule

“…Therefore the three solutions are not to be mixed and stay as separate solutions. They all satisfy (2). The formulas (16) for G x = 0 and (21) for G x = 0 give all solutions of (2).…”

“…Conversely, any operator T of the form (4) together with A = A 1 = A 2 of the form (3) satisfies (2).…”

“…Suppose that (T , A 1 , A 2 ) satisfy the functional equation (2), with (T , A 1 ) nondegenerate and (A 1 , A 2 ) pointwise continuous. Assume further that for some y ∈ R there is g ∈ C 1 (R) with g(y) = 0 and T g(y) = 0.…”