Lectures on functional equations and their applications

Aczel J.350-359-412-4

The solution of functional equations is one of the oldest topics of mathematical analysis. D’Alembert, Euler, Gauss, Cauchy, Abel, Weierstrass, Darboux, and Hilbert are among the great mathematicians who have been concerned with functional equations and methods of solving them. In this field of mathematics, as in others, the literature has grown markedly during the past fifty years. (See the chronological bibliography at the end of this volume.) However, results found in earlier decades have often been presented anew because through the years there has been no systematic presentation of this field, in spite of its age and its importance in application. In this monograph, an attempt is made to remedy this situation, at least in part. Results are usually presented with proofs, in contrast to S. Pincherle’s German and French encyclopedia articles published in 1906 and 1912, which, of course, were written for a different purpose. Earlier works (such as those by E. Czuber 1891, E. Picard 1928, G. H. Hardy, J. E. Littlewood, and G. Polya 1934, M. Frechet 1938, and B. Hostinsky 1939) (see bibliography) also give some attention to functional equations, but the special functional equations treated are subordinate to their applications. We prefer to arrange the subject matter according to actual types of functional equations. We also cover a different and, as we think, somewhat broader range of problems than does the book of M. Ghermanescu 1960[b]. A. R. Schweitzer’s plan of 1918 to compile a bibliography of the theory of functional equations was, alas, never carried out; therefore the list of references at the end of this book, although incomplete, can partly serve as a bibliography too.