A Panorama of Hungarian Mathematics in the Twentieth Century: Non-Commutative Harmonic Analysis

For present purposes, we shall define non-commutative harmonic analysis to mean the decomposition of functions on a locally compact G-space X, where G is some (locally compact) group, into functions well-behaved with respect to the action of G. The classical cases are of course Fourier series, when G = X = T, the circle group, and the Fourier transform, when G = X = R, but we will mostly be concerned with the case when G is non-commutative. Since this subject is inextricably linked with the subject of representations of G (unitary representations, if we specialize to the case of L-functions), we will also consider the general theory of representations of locally compact groups and of various related structures, such as Lie algebras and Jordan algebras. The subject of group representations was created by Georg Frobenius [9] in a remarkable series of papers in the 1890’s, and continued in the first decade of the twentieth century in the work of his student Issai Schur [19]. However, Frobenius worked exclusively with finite groups, and his treatment was purely algebraic. It took a while before it was realized that Frobenius’ theory had important implications for harmonic analysis. The generalization of the theory to compact groups was largely carried out by Hermann Weyl, and applications to harmonic analysis on compact groups did not come until the Peter-Weyl Theorem ([17]; reprinted in [31], pp. 387–404). It is against this background that we shall consider the contributions of a few great Hungarian mathematicians: Alfred Haar, John von Neumann, and Eugene Wigner in the 1920’s, 1930’s, and 1940’s; and in somewhat later generations, Béla Szőkefalvi-Nagy and Lajos Pukánszky. As there is room here to discuss only a few of their contributions, we refer the reader to the scientific obituaries [20], [25], [16], [10], [15], [33], [26], [6], and [5] for more details.