This is a review of Collected Works of John Tate. Parts I, II, edited by Barry Mazur and Jean-Pierre Serre. American Mathematical Society, Providence, Rhode Island, 2016. For several decades it has been clear to the friends and colleagues of John Tate that a “Collected Works” was merited. The award of the Abel Prize to Tate in 2010 added impetus, and finally, in Tate’s ninety-second year we have these two magnificent volumes, edited by Barry Mazur and Jean-Pierre Serre. Beyond Tate’s published articles, they include five unpublished articles and a selection of his letters, most accompanied by Tate’s comments, and a collection of photographs of Tate. For an overview of Tate’s work, the editors refer the reader to [4]. Before discussing the volumes, I describe some of Tate’s work. 1. Hecke L-series and Tate’s thesis Like many budding number theorists, Tate’s favorite theorem when young was Gauss’s law of quadratic reciprocity. When he arrived at Princeton as a graduate student in 1946, he was fortunate to find there the person, Emil Artin, who had discovered the most general reciprocity law, so solving Hilbert’s ninth problem. By 1920, the German school of algebraic number theorists (Hilbert, Weber, . . .) together with its brilliant student Takagi had succeeded in classifying the abelian extensions of a number field K: to each group I of ideal classes in K, there is attached an extension L of K (the class field of I); the group I determines the arithmetic of the extension L/K, and the Galois group of L/K is isomorphic to I. Artin’s contribution was to prove (in 1927) that there is a natural isomorphism from I to the Galois group of L/K. When the base field contains an appropriate root of 1, Artin’s isomorphism gives a reciprocity law, and all possible reciprocity laws arise this way. In the 1930s, Chevalley reworked abelian class field theory. In particular, he replaced “ideals” with his “idèles” which greatly clarified the relation between the local and global aspects of the theory. For his thesis, Artin suggested that Tate do the same for Hecke L-series. When Hecke proved that the abelian L-functions of number fields (generalizations of Dirichlet’s L-functions) have an analytic continuation throughout the plane with a functional equation of the expected type, he saw that his methods applied even to a new kind of L-function, now named after him. Once Tate had developed his harmonic analysis of local fields and of the idèle group, he was able prove analytic continuation and functional equations for all the relevant L-series without Hecke’s complicated theta-formulas. Received by the editors September 5, 2016. 2010 Mathematics Subject Classification. Primary 01A75, 11-06, 14-06. c ©2017 American Mathematical Society

The collected works

Zeros of orthogonal polynomials

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

Introduction 1. Packaging and covering densities 2. The existence of reasonably dense packings 3. The existence of reasonably economical coverings 4. The existence of reasonably dense lattice packings 5. The existence of reasonably economical lattice coverings 6. Packings of simplices cannot be very dense 8. Coverings with spheres cannot be very economical Bibliography Index.

Packing and Covering. By C. A. Rogers. Pp. viii, 111. 30s. 1964. (Cambridge)

Hyperbolic cross approximation is a special type of multivariate approximation. Recently, driven by applications in engineering, biology, medicine and other areas of science new challenging problems have appeared. The common feature of these problems is high dimensions. We present here a survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions. The theory of hyperbolic cross approximation and related theory of functions with mixed smoothness are under detailed study for more than 50 years. It is now well understood that this theory is important both for theoretical study and for practical applications. It is also understood that both theoretical analysis and construction of practical algorithms are very difficult problems. This explains why many fundamental problems in this area are still unsolved. Only a few survey papers and monographs on the topic are published. This and recently discovered deep connections between the hyperbolic cross approximation (and related sparse grids) and other areas of mathematics such as probability, discrepancy, and numerical integration motivated us to write this survey. We try to put emphases on the development of ideas and methods rather than list all the known results in the area. We formulate many problems, which, to our knowledge, are open problems. We also include some very recent results on the topic, which sometimes highlight new interesting directions of research. We hope that this survey will stimulate further active research in this fascinating and challenging area of approximation theory and numerical analysis.

Hyperbolic Cross Approximation

Trigonometric series with general monotone coefficients

Ul'yanov and Nikol'skii-type inequalities

In this paper we investigate properties of the general monotone sequences and functions, a generalization of monotone sequences and functions as well as of those of bounded variation. Some applications to various problems of analysis are given. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Sergey Tikhonov

Papers

Trigonometric series with general monotone coefficients

Ul'yanov and Nikol'skii-type inequalities

A concept of general monotonicity and applications

Best approximation and moduli of smoothness: Computation and equivalence theorems

Sharp Jackson inequalities