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This book is about regularity properties of functional equations. In the second part of his fifth problem, Hilbert asked, concerning functional equations, "In how far are the assertions which we can make in the case of differentiable functions true under proper modifications without this assumption?" This book contains, in a unified fashion, most of the modern results about regularity of non-composite functional equations with several variables. These results show that "weak" regularity properties, say measurability or continuity, of solutions imply that they are in C[infinity], and hence the equation can be reduced to a differential equation. A long introduction highlights the basic ideas for beginners. Several applications are also included.
Audience
This book is intended for researchers in the fields of mathematical analysis, applied mathematics, theoretical economics, and statistics.
List of contents
Preliminaries.- Steinhaus Type Theorems.- Boundedness and Continuity of Solutions.- Differentiability and Analyticity.- Regularity Theorems on Manifolds.- Regularity Results with Fewer Variables.- Applications.
Summary
This book illustrates the basic ideas of regularity properties of functional equations by simple examples. It then treats most of the modern results about regularity of non-composite functional equations of several variables in a unified fashion. A long introduction highlights the basic ideas for beginners and several applications are also included.
Additional text
From the reviews of the first edition:
"This book is a welcome addition to the functional equations literature. … The author does a masterful job of organizing and summarizing the results of the last 50 years and weaving them together into a coherent general theory. There are over 200 references. The book is very carefully written … . The author has certainly achieved his stated goal of making these results ‘more accessible and easier to use for everyone working with functional equations.’" (Bruce Ebanks, Zentralblatt MATH, Vol. 1081, 2006)
Report
From the reviews of the first edition:
"This book is a welcome addition to the functional equations literature. ... The author does a masterful job of organizing and summarizing the results of the last 50 years and weaving them together into a coherent general theory. There are over 200 references. The book is very carefully written ... . The author has certainly achieved his stated goal of making these results 'more accessible and easier to use for everyone working with functional equations.'" (Bruce Ebanks, Zentralblatt MATH, Vol. 1081, 2006)
