The book "New methods of approximation in the problems of actual analysis and in spectral theory", written by the author's team headed by Khromov A. P. and Lukomsky S. F. , is a unique study that opens up new horizons in the field of mathematical analysis and spectrum theory. This edition will be of particular interest to those seeking to deepen their knowledge in mathematics, as well as to practitioners working in related fields such as physics, engineering, and computer science. The book discusses modern methods of approximation, which find their application in solving complex problems of real analysis. The authors offer the reader not only theoretical foundations, but also practical examples, which makes the material accessible and understandable. The book covers a wide range of topics, ranging from classical approaches to the latest developments, which allows the reader to see the evolution of approximation methods and their application in various contexts. One of the key topics raised in the book is the relationship between approximation and spectral theory. The authors analyze in detail how different approximation methods can be used to study operator spectra, which is an important task in mathematical physics. This makes the book particularly valuable for undergraduate and graduate students, as well as for researchers working in theoretical physics and applied mathematics. The book will be of interest not only to specialists, but also to those who are just starting their way in the world of mathematics. The style of presentation of the authors is clear and logical, which makes it easy to assimilate complex concepts. The use of illustrative examples and tasks contributes to a better understanding of the material, and also helps the reader to apply the knowledge gained in practice. For those interested in mathematical analysis and spectral theory, this book will be an indispensable tool. It can be useful for both senior students and teachers who want to update their knowledge and teaching methods. In addition, it will be of interest to researchers working in the field of numerical methods and operator theory, providing new ideas and approaches for their work. Authors, A. P. Chroms and C. F. Lukomsky is known in scientific circles for his research in the field of mathematical analysis and function theory. Their previous works have already received recognition for their originality and depth of analysis. In this book, they continue to develop their ideas, offering the reader a fresh look at established concepts and methods. “New methods of approximation in real analysis and spectral theory” is not just a textbook, but a real find for anyone who wants to understand the intricacies of mathematical analysis and its applications. The book will be an excellent addition to the library of both students and professionals seeking new knowledge and discoveries in the world of mathematics. If you are looking for literature on mathematical analysis, spectral theory or numerical methods, this edition is definitely worth your attention. Discover new horizons of mathematics with this fascinating and informative book!