《C.Fefferman 在1974年数学家大会上的报告》.pdf

Proceedings of the International Congress of Mathematicians Vancouver, 1974 Recent Progress in Classical Fourier Analysis Charles Fefferman* ix n In what sense does J „ e *f(g) d£ converge to a given function/ on R l How do Ä properties such as the size and smoothness of/influence the behavior of its Fourier transform/? These simple questions lie at the heart of much of classical analysis. Their deeper study leads naturally to certain basic auxiliary operators defined on n functions on R ; and Fourier analysts seek to understand these operators and their generalizations, and to apply them to various branches of analysis. In this paper I shall describe some basic results and applications of Fourier analysis and speculate briefly on the future. I have left out many topics of great importance, and empha- sized merely those subjects I know something about. Let me begin by sketching the state of the art as of about 1950. At that time, the field was well developed only in the one-dimensional case. Since it had long been known that the Fourier series of a continuous function on[0, 2%\ need not converge at every point, Lebesgue measure (and in particular U) was clearly recognized as 2 2 a basic tool. The Plancherel theorem ffi\f(x)\ dx = 2% 2 ^ \a \ with/(x) ~ k eihx a 2 J^ooCik 8 ve a complete characterization of L functions in terms of their Fourier coefficients and established norm convergenc