During the spring semester of 1979 we presented a program in Euclidean harmonic analysis at the University of Maryland. The six lecture series comprising this volume were a major part of our program. Euclidean harmonic analysis has a rich basic theory and maintains a vital relationship with several other areas which, in fact, have molded the subject and enlivened it with significant applications for over 150 years. Wiener's Tauberian theorem provides a neat example of this fundamental and, to some extent, mysterious interplay. Wiener's theorem not only characterizes the prime number theorem but is used to define spectra properly for phenomena such as white light; this spectral theory provides perspective for the Fourier analysis associated with correlation functions in filtering and prediction problems, and these problems, in turn, lead naturally to H p spaces. In the first lecture series of this volume L. CARLESON addressed the two main problems of classical statistical meehanies: a. the verification of expected equilibrium thermodymamic properties and b. the validity of the Gibbs theory for dynamical systems. The results of part a include proofs of the basic properties of the free energy function, as well as a rigorous verification of the existence of phase transition for certain classical models. In part b Carleson first discusses a Boltzmann equation and the approach to equilibrium that it describes. He then considers dynamical properties of harmonic oscillator systems and shows how one can verify the Gibbs theory for an ensemble of such systems. Classical harmonic analysis is pervasive in his approach; and the point of his lectures is to introduce some analytic results and problems which may eventually lead to further progress in applications. The remaining lecture series contained in this volume, as well as the lectures by our other visitors, fell into one or the other of two categories of problems.
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