Cambridge University Press publishes Single and Multiple Number Series, co-authored by Sergey Tikhonov (ICREA, CRM)

Cambridge University Press has published Single and Multiple Number Series, volume 189 of its Encyclopedia of Mathematics and Its Applications. The book is the work of four authors: Alberto Debernardi Pinos (Universitat Autònoma de Barcelona), Elijah Liflyand (Bar-Ilan University), Sergey Tikhonov (ICREA, CRM) and Maria Zeltser (Tallinn University).

The theory of number series sits at the base of much of analysis. Deciding whether an infinite sum settles on a finite value or grows without bound is a question with two centuries of history behind it, and most of the classical convergence tests, from Cauchy’s condensation test to the integral test, answer it under one standing assumption: that the terms of the series decrease in an orderly, monotone way. In practice, that assumption often fails. The coefficient sequences that arise in Fourier analysis, for instance, tend to oscillate as they decay, and the classical tests, read strictly, do not apply to them.

A central line of the book examines how far this monotonicity condition can be relaxed while the classical results remain true. The authors work with two weaker notions. Weak monotonicity allows a term to exceed its neighbours within a fixed factor, and turns out to be enough for several of the recognised tests to survive. Where it does not suffice, the book turns to general monotone sequences, a class Tikhonov introduced in the mid-2000s that has since become a standard tool in Fourier and functional analysis. Rather than the size of the terms, the condition controls the total variation of the sequence against the magnitude of its own terms. Under these relaxed conditions, many convergence and divergence tests apply to a considerably wider set of series than the ones they were originally written for.

The second half of the book moves to higher dimensions. For double and multiple series the notion of convergence itself is no longer unique, since the partial sums can be ordered in several ways that do not always agree, a difficulty first examined systematically by Pringsheim around 1900. The volume treats the two-dimensional case in a survey form and then develops the theory in arbitrary dimension with full proofs. According to the authors, it is the first detailed book-length presentation of the theory of multiple number series.

The text is organised in three parts of three chapters, covering single, double and multiple series in turn. Each chapter closes with historical notes tracing the origin of its results, and the volume includes a table of correspondences between the one-dimensional theorems and their higher-dimensional counterparts. Power series and functional series are deliberately left outside its scope. The book is addressed to a wide readership, from undergraduate students to specialists in the field.

Sergey Tikhonov has been an ICREA Research Professor at the CRM since 2012. His research areas are Fourier analysis and approximation theory.

Single and Multiple Number Series. Alberto Debernardi Pinos, Elijah Liflyand, Sergey Tikhonov and Maria Zeltser. Encyclopedia of Mathematics and Its Applications 189, Cambridge University Press, 2026. DOI: 10.1017/9781009562973.