This monograph develops an operator viewpoint for functional equations
in classical function spaces of analysis, thus filling a void in the
mathematical literature. Major constructions or operations in analysis
are often characterized by some elementary properties, relations or
equations which they satisfy. The authors present recent results on the
problem to what extent the derivative is characterized by equations such
as the Leibniz rule or the Chain rule operator equation in Ck-spaces. By
localization, these operator equations turn into specific functional
equations which the authors then solve. The second derivative,
Sturm-Liouville operators and the Laplacian motivate the study of
certain "second-order" operator equations. Additionally, the authors
determine the general solution of these operator equations under weak
assumptions of non-degeneration. In their approach, operators are not
required to be linear, and the authors also try to avoid continuity
conditions. The Leibniz rule, the Chain rule and its extensions turn out
to be stable under perturbations and relaxations of assumptions on the
form of the operators. The results yield an algebraic understanding of
first- and second-order differential operators. Because the authors have
chosen to characterize the derivative by algebraic relations, the rich
operator-type structure behind the fundamental notion of the derivative
and its relatives in analysis is discovered and explored.
The book does not require any specific knowledge of functional
equations. All needed results are presented and proven and the book is
addressed to a general mathematical audience.
