*Approximate Methods of Higher Analysis* by L.V. Kantorovich and V.I. Krylov is a reprint by Dover of the 1958 English translation of the third edition of the Russian text from 1950. The original Russian edition was published in 1936 with a second edition appearing in 1941.

At the time of its publication, *Approximate Methods of Higher Analysis* covered the state of the art in the approximation theory of solutions of partial differential and integral equations. In the English translation of the third edition, the bibliography contains references spanning from about 1870 to about 1950, with most of the references being roughly within the range of 1935 - 1945. Further, many of the references are to original research articles from Russian journals that were not easily accessible to much of the world for a significant period of time. Thus, it seems that the appearance of the English translation of the book opened up the research literature in applied analysis.

*Approximate Methods of Higher Analysis* is devoted to methods of approximation theory of boundary value problems of the type that commonly arise in classical mathematical physics. Specifically, problems in the study of the gravitational potential, electrostatics, waves, heat conduction, and continuum mechanics lead to boundary value problems for differential or integral equations. These problems had a profound impact on some directions of research in mathematical analysis in the late nineteenth and early twentieth centuries, an influence that persists through to this day. First, there are problems in developing mathematically rigorous methods for understanding the general properties of solutions to boundary value problems. The requirements of engineering and physics naturally lead to methods of approximation of solutions to boundary values problems in cases where obtaining exact solutions is intractable. This leads to the second type of problem, the rigorous mathematical analysis of properties, e.g. convergence and stability, of these methods of approximation. This is what *Approximate Methods of Higher Analysis* is all about.

Kantorvich and Krylov begin their book by using infinite series solutions to obtain (via truncation) approximate solutions to the two most common boundary value problems for Laplace's equation. Specifically, they employ the method of separation of variables and Fourier series. Next, they extend this approach via more general orthogonal function expansions (there is a nice presentation of the Gram-Schmidt procedure) and then generalize to certain non-orthogonal expansions. Non-orthogonal function expansions lead to problems that require solving (countably) infinite linear systems. Thus, Kantorvich and Krylov develop the necessary theory of infinite systems of linear equations. Here the reader can see some of the early stages of what has become the more general theory of linear functional analysis. In fact, there are many places in the text where the reader can see the early notions of applied functional analysis and I think it is worth noting that L.V. Kantorovich (awarded the Nobel Prize in Economics) made substantial contributions to functional analysis. *Approximate Methods of Higher Analysis* proceeds to develop the theory of approximation of solutions to integral equations by the method of successive approximation, finite-difference approximation of solutions to differential (ordinary and partial) equations, variational methods (the beginnings of the mathematical theory of finite-element methods) for the approximation of solutions to differential equations, and conformal mapping methods and Schwarz's method for handling domains with geometry more complicated than that of a rectangle or circle.

While the methods studied in

*Approximate Methods of Higher Analysis* are not strictly numerical methods (i.e. they are not numerical algorithms) much of what is covered in the book would now fall under the umbrella of modern numerical analysis, and many of the techniques, e.g. finite-difference methods, do lead to numerical algorithms. Great strides have been made in the field of numerical analysis both in terms of theory and practical application (especially the computer implementation of methods) since the publication of

*Approximate Methods of Higher Analysis*. Nevertheless, the book remains influential and continues to be referenced in the research literature. However, I would not recommend

*Approximate Methods of Higher Analysis* as a starting point for learning approximation theory or methods such as finite-difference and finite-element methods. The notation is somewhat antiquated and there are many important concepts and results that one should know that were only developed its publication. An excellent modern text that follows the legacy of

*Approximate Methods of Higher Analysis* is the third edition of

*Theoretical Numerical Analysis* by Atkinson and Han (which even cites Kantorvich and Krylov.)

In conclusion, I want to briefly consider some views of *Approximate Methods of Higher Analysis* both in terms of the reception of the book at the time it was published and also it's legacy. After the English translation of the third edition appeared, the book was reviewed by Wilfred Kaplan in the May 1960 Bulletin of the AMS. Kaplan refers to the book as ``remarkable'' and writes that the book ``is highly to be commended''. In his article, ``From finite differences to finite elements: A short history of numerical analysis of partial differential equations'' Vidar Thomèe cites *Approximate Methods of Higher Analysis* as a historical reference for the development of the idea of ``using a variational formulation of a boundary value problem for its numerical solution''. Finally, in the article ``Correcting Three Errors in Kantorivich & Krylov's *Approximate Methods of Higher Analysis*'' from the March 2016 American Mathematical Monthly, John P. Boyd demonstrates how computer algebra systems can be employed to improve on classical results obtained by hand in *Approximate Methods of Higher Analysis*. Thus we see evidence that this book was important in its time, that it remains an important historical reference, and that fruitful problems may still be derived from its pages. So while *Approximate Methods of Higher Analysis* may not be the in demand reference text it once was, there is still plenty of good reason to browse through the book, especially if you have any interest at all in the subjects that it covers.

Jason Graham is an Associate Professor in the Department of Mathematics at the University of Scranton. He received his PhD from the program in Applied Mathematical and Computational Sciences at the University of Iowa. His professional interests are in applied mathematics and mathematical biology.