Convergence Problems of Orthogonal Series deals with the theory of convergence and summation of the general orthogonal series in relation to the general theory and classical expansions. The book reviews orthogonality, orthogonalisation, series of orthogonal functions, complete orthogonal systems, and the Riesz-Fisher theorem. The text examines Jacobi polynomials, Haar's orthogonal system, and relations to the theory of probability using Rademacher's and Walsh's orthogonal systems. The book also investigates the convergence behaviour of orthogonal series by methods belonging to the general theory of series. The text explains some Tauberian theorems and the classical Abel transform of the partial sums of a series which the investigator can use in the theory of orthogonal series. The book examines the importance of the Lebesgue functions for convergence problems, the generalisation of the Walsh series, the order of magnitude of the Lebesgue functions, and the Lebesgue functions of the Cesaro summation. The text also deals with classical convergence problems in which general orthogonal series have limited significance as orthogonal expansions react upon the structural properties of the expanded function. This reaction happens under special assumptions concerning the orthogonal system in whose functions the expansion proceeds. The book can prove beneficial to mathematicians, students, or professor of calculus and advanced mathematics.

The text of this book is an improved and extended version of the German original, for, since the issue of the latter many interesting results were published which I have thought necessary to include in the text. At the same time, I have corrected some errors and misprints of the original text. In this task I have derived much assistance from the valuable remarks of Ákos Császár and Géza Freud. I am particularly indebted to Károly Tandori for having revised the complete English text and read its proof-sheets. Finally, I express my gratitude to Imre Földer for the careful translation.

The questions of convergence and summation of the general orthogonal series forms perhaps the most impressive domain of application of the Lebesgue or of the Stieltjes-Lebesgue concept of integral, respectively. Many methods of inquiry owe their discovery to the investigation of this sphere of problems. In spite of their great generality, some of the results obtained provide wider knowledge of the convergence features than the remaining theorems, shaped specially to the expansion in question, even in case of applications to classical orthogonal expansions. Thus, for instance, the Menchoff-Rademacher convergence theorem for general orthogonal series ensures the convergence almost everywhere of certain Fourier series with irregularly distributed lacunarities, while the special theorems achieved up to the present time on the convergence of the Fourier series are unable to answer this question in such cases. Furthermore, the convergence problems of the orthogonal series are very tightly bound up with some other branches of Analysis, especially with probability theory. It may even be stated that a set of theorems from the theory of orthogonal series and from the theory of probability are basically only bilingual terms for the same mathematical fact.

The large range and the depth of the convergence theory of the orthogonal series justifies a systematic treatment of this theory. Although a programme of such a kind was excellently carried out in the well-known book of Kaczmarz and Steinhaus, the zeal of the mathematicians has opened a way to new, beautiful and important discoveries during the 25 years which have elapsed since its publication. In view of this circumstance, it seems to me reasonable to hope that this book will not call forth the sentiment that it is superfluous.

I have attempted to represent the actual state of the theory of convergence and summation of the general orthogonal series with hints as to the connexion of the general theory with the corresponding questions of the classical expansions. On the other hand, however, I have not dealt with other important ranges of ideas, unrelated to questions of convergence, as for instance those connected with the theorem of Young-Hausdorff or the theorem of Paley.

I endeavoured to formulate the text in such a way that any reader, acquainted with the most important facts from the theory of functions of a real variable and from the theory of the Fourier series, will find all the rest completely proved in this book, excepting only the parts printed in smaller type, containing various topics: theorems with complete or with only sketched proof or even without proof, references to unsolved problems, remarks for the better classification of the main text, etc. I have also striven to give a hint of the origin of the several theorems, not only by indicating the place where the theorem in question has been formulated for the first time in its most general form, but frequently also by referring to the older literature in which the fundamental idea of the proof has first appeared.

I am very much indebted, relative to the form as well as to the content, to the monograph of Kaczmarz and Steinhaus, to the book of Szegö on orthogonal polynomials and to the well-known work of Zygmund on trigonometrical series whose new, greatly extended edition, however, could unfortunately not actually be utilized for this book. The appendix of Guter and Ulianoff provided for the Russian translation of the book of Kaczmarz and Steinhaus (Moscow, 1958) has likewise been very useful for our purposes.

I avail myself of the opportunity to express my deepest gratitude to my colleagues B Sz.-Nagy and K Tandori for their generous help and very valuable comments during the writing of this book. Their remarks, advice and assistance have contributed appreciably to the improvement of the text. I have also to thank to the publishing house of the Hungarian Academy of Sciences, as well as to the printing office of Szeged, for their careful production of the book.

4.1. Chapter I - Fundamental Ideas. Examples of Series of Orthogonal Functions.

This chapter discusses orthogonality, orthogonalisation, and series of orthogonal functions. The chapter presents the notion of orthogonality by means of the Stieltjes-Lebesgue integral. The Riesz-Fischer theorem is discussed. The chapter discusses the completeness of an orthogonal system, and the completeness of the trigonometrical system. Orthogonal polynomials, and the Christoffel-Darboux formula are discussed. The chapter reviews convergence theorem for expansions in orthogonal polynomials. The Christoffel-Darboux formula indicates a certain similarity of expansions in orthogonal polynomials with Fourier expansions. From this formula it is possible to deduce a convergence theorem allowing several applications and having a classical analogue in the theory of Fourier series. The Jacobi polynomials are discussed. Among all the Jacobi polynomials, the Chebysheff polynomials are perhaps of the greatest importance. The chapter also reviews the bounds for general orthonormal systems and orthonormal systems of polynomials, and discusses the Haar's orthogonal system.

4.2. Chapter II - Investigation of the Convergence Behaviour of Orthogonal Series by Methods Belonging to the General Theory of Series.

This chapter discusses the convergence behaviour of orthogonal series by methods belonging to the general theory of series. The chapter reviews the Abel-Poisson summation process, and the Abel transform. The classical Abel transform of the partial sums of a series is very useful in the theory of orthogonal series. In the theory of series, those theorems are called Tauberian that afford criteria enabling one to conclude the convergence of a weaker summation process from the convergence of a stronger one. The fundamental theorem concerning the convergence of orthogonal series is discussed. Generalities on the Cesàro summation of orthogonal series are reviewed. The chapter presents the coefficient tests for the Cesàro summability of orthogonal series, and reviews the summability of lacunary orthogonal series. Riesz summation of orthogonal series is reviewed. The chapter also discusses Abel non-summable orthogonal series with monotone coefficients, and the Menchoff's summation theorem.

4.3. Chapter III - The Lebesgue Functions.

This chapter discusses the significance of the Lebesgue functions for convergence problems. The chapter reviews a convergence condition, based on the order of magnitude of the Lebesgue functions. The Fourier series and the orthogonal polynomial expansions exhibit much similarity concerning their convergence properties. The chapter discusses the multiplicatively orthogonal systems, and presents the generalisation of the Walsh series. Convergence of strongly lacunary Fourier series is discussed. The chapter discusses the Lebesgue functions of the Cesàro summation. Summability of orthogonal series arising in terms of the functions of a polynomial-like system is discussed. The order of magnitude of the Lebesgue functions is discussed. The chapter also reviews the impossibility of the improvement of the coefficient tests.

4.4. Chapter IV - Classical Convergence Problems.

This chapter discusses the classical convergence problems. The classical convergence questions refer to the representability of a function with given structure at single points, that is, it is asked which are those structural properties of a function $f(x)$ that secure the convergence or, respectively, the summability of its orthogonal expansion at the point $x_{0}$ given in advance, the sum having the value $f(x_{0})$. The chapter discusses under which conditions the convergence, provided that it takes place, is uniform in certain intervals and how quickly the function $f(x)$ is approximated by its orthogonal expansion. The chapter discusses some fundamental conceptions and facts of functional analysis and establishes from them the necessity of several assumptions. A set is said to be of the first category if it is the sum of a denumerable set of nowhere dense sets. In the opposite case, it is said to be of the second category. A subset of the second category of a Banach space is non-denumerable, as a set consisting of a single point being nowhere dense, a denumerable set is of the first category.

The standard text on the theory of general orthogonal series is that by S Kaczmarz and H Steinhaus (Warsaw, 1935). One of the main features of their account is the systematic use of ideas and methods of functional analysis then coming to the fore, and which are, indeed, indispensable for a structural unity of the theory. Naturally, in the 25 years since then much progress has been made in this field. In their turn, there are now the more refined methods of classical analysis which have contributed most to this progress. In particular, such methods have been developed more recently in Hungary and Russia. The new book by one of the leading Hungarian analysts in this field gives an excellent and up to date account of this development. It thus supplements the older book in many important directions without superseding it. There is, of course, some overlapping in content, but the new book aims mainly at a thorough account of the summability problems for orthogonal series in general, and of orthogonal Fourier series in the class $L^{2}_{\mu}$ (over a finite closed real interval) in particular. The ideas and methods are those of classical analysis. Only in the last chapter where the theory of convergence for the class $L_{\mu}$ is discussed, some use of the "condensation principle" of functional analysis (crucial in the Kaczmarz-Steinhaus text) is made. Thus the Hausdorff-Young theory for the general classes $L^{p}_{\mu}$ is deliberately omitted. The author gains in this way a self-contained approach on classical lines to his selected problems.

The book is admirably written. The author not only produces readable rigorous proofs, but these are often simpler than those in the original papers. He takes much pains in making the underlying ideas of the often complicated techniques clear, he provides ample references and poses many interesting conjectures and problems. I have no doubt that his book will remain, for a long time to come, a second standard text for anyone who wants to study seriously this fascinating and important subject.

The printing is excellent, and there are not many misprints, none of them serious.

The book is a monograph, which presents (mainly) questions related to the convergence or summability of orthogonal series. On the one hand, the book presents already widely known (old) results, and on the other hand, it contains many new results obtained in recent years and published in various mathematical journals. In view of this, Alexits' book reflects the current state of the theory of orthogonal series in questions of convergence and summability, and therefore is (in a sense) a continuation of the book by Kaczmarz and Steinhaus "The Theory of Orthogonal Series" (Moscow, 1958), which is devoted to the general theory of orthogonal series and which was written in 1935. It is useful to note that Alexits' book contains many digressions in which the author provides historical background and also points out a number of unsolved problems.