欧拉视角下的正交多项式与连分数：18世纪数学巨匠的融合

2008年

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"《正交多项式与连分数：欧拉视角》是一部由塞尔日·克鲁什切夫教授编著的数学著作，于2008年7月24日由剑桥大学出版社出版。该书聚焦在十八世纪数学巨匠欧拉对这两个主题——正交多项式和连分数——的开创性贡献。连分数，自古希腊时期就被研究，但在欧拉的手中，它们成为了一种强大的工具。书中详述了欧拉如何引入正交多项式的概念，并将两者紧密结合，从而推进了特殊函数和正交多项式的理论发展。尤为关键的是，书中探讨了欧拉工作的一些重要应用，如马克霍夫定理关于拉格朗日谱的阐述，阿贝尔积分在有限项下的证明，切比雪夫关于正交多项式的理论，以及最近在单位圆上正交多项式的最新进展。连分数的重要性再次被提升，部分归功于其在逼近理论中的算法寻找中的应用。因此，这本书适时地复兴了沃尔利斯、布劳恩科克和欧拉的传统方法，展示了他们的影响力在当今数学研究中的持续意义。特别值得一提的是，本书附录收录了欧拉著名的论文《连分数，观察》，这是对这位数学巨匠思想的一次珍贵展示。作为阿提里姆大学数学系的教授，克鲁什切夫不仅提供了深入的学术分析，还通过对历史和现代应用的交织，向读者展示了数学理论如何随着时间的推移而发展和演变。整本书不仅是对欧拉遗产的一次深入研究，也是对数学史上的两个重要分支——正交多项式和连分数——的一次全面回顾。"

Preface xv

makes the basic ideas of Markoff look more natural. With Jean Bernoulli sequences

and the formulas for Jean Bernoulli and Markoff periods one can easily calculate

numerically as many points of the Lagrange spectrum as necessary.

Markoff’s theory completes the algebraic part of this book. The analytic part begins

in Chapter 3 and is followed in Chapter 4 by Euler’s research. To a great extent

this chapter covers Euler (1750b) but with the difference that Brouncker’s method

recovered in Chapter 3 is applied. The method can be extended from the unit circle

considered by Wallis to the class of sinusoidal spirals introduced into mathematics

in 1718 by another great British mathematician, Colin Maclaurin. Chapter 4 covers

the forgotten Euler differential method of summation of some continued fractions of

hypergeometric type. It turns out that, for instance, approximately half the continued

fractions discovered later by Stieltjes can be easily summed up by this method of

Euler. However, the differential method does have some limitations. Attempting to

overcome them Euler arrived at the beautiful theory of Riccati equations. In (1933)

Sanielevici presented Euler’s method in a very general form. Later Khovanskii (1958)

using Sanielevici’s results developed as continued fractions many elementary functions.

Still, I think that Euler’s method as stated by Euler makes everything more clear. In

this part I filled some gaps in the proofs while trying not to violate Euler’s arguments.

The central result here is the continued fraction for the hyperbolic cotangent. I collect

in Chapter 5 some results which were or could be directly or indirectly influenced by

Euler’s formulas. Such an approach sheds new light on the subject.

Chapter 6 presents results either obtained by Wallis interpolation or by a direct

transfer from the regular continued fractions of number theory to polynomial con-

tinued fractions, i.e. to P-fractions. Euler’s results on hypergeometric functions play

a significant role here. Another interesting topic of this chapter is the periodicity

of P-fractions. As before, the first results were obtained by Euler. Using contin-

ued fractions of the radicals of quadratic polynomials as guidance, Euler found his

now well-known substitutions for integration. This was extended by Abel in one

of his first papers, which incidentally preceded his discoveries in elliptic functions.

I include in this chapter a beautiful result of Chebyshev on integration in finite

terms.

Chapter 7 indicates how Euler’s ideas eventually led to the discovery of orthogonal

polynomials. Finally, I present in Chapter 8 my own research on the convergence of

Schur’s algorithm.

A few words on the title. It varied several times but since the essential part of the

book is related to Euler I believe that finally I made a good choice. Moreover, in 2007

the tercentenary of Euler was celebrated.

Following Euler, I have split the book into small numbered “paragraphs” (sub-

sections). This was an old tradition in mathematics, now almost forgotten. It makes

the book easier to read. The difference from what Euler did is that almost all the

“paragraphs” also have titles.

xvi Preface

This book is not a complete account of what has been done in orthogonal polynomials

or in continued fractions. In orthogonal polynomials Szeg

o’s book (1939) is still

important. There are also two important contributions made by Nevai (1979, 1986),

and another two books by Saff and Totik (1997) and by Stahl and Totik (1992). As to

orthogonal polynomials on the unit circle there is the recent and exhaustive work in

two volumes of Barry Simon (2005). More on continued fractions can be found in

Jones and Thron (1980), Khinchin (1935), Khovanskii (1958), Perron (1954, 1957)

and Wall (1943).

Most parts of this book require only some knowledge of calculus and an under-

graduate course on algebra. In Chapters 6–8 elementary facts from complex analysis

are used occasionally. In Chapter 8 in addition it is important to know basic facts on

Hardy spaces. There are two relatively new and very well written books on Hardy

spaces: Garnett (1982) and Koosis (1998).

I wish to thank a number of people and organizations supporting me in one or

another way during the work on this project. First of all I express deep gratitude to my

aunt Galina Kreidtner. She was not a mathematician, she was an architect and artist.

Nonetheless she enthusiastically and helpfully discussed with me the idea of this book

in Almaty. She died at the age of 80 in 2000 and so I lost one of the best friends in

my life. I dedicate this book to her memory.

Another very good friend, Paul Nevai from the Ohio State University, made a right

choice in favor of orthogonal polynomials at the very beginning of his mathematical

career in St Petersburg University, where we were fellow students. His support was

also extremely valuable and sincere.

My special thanks to Purdue University, Indiana, which played a significant role in

my career in mathematics at least twice. I am particularly grateful to David Drasin

and Carl Cowen.

I am very grateful also to Atilim University, Ankara, which has created very good

conditions for my research in mathematics and supported all my scientific projects.

I express my sincere and deep gratitude to Barry Simon (Caltech, Pasadena) who

provided very important personal support for this book.

The book turned out to be so much influenced by Wallis’ Arithmetica Infinitorum

(1656) that I cannot avoid a temptation to finish this preface with a citation from its

very end:

There remains this: we beseech the skilled in these things, that what we thought worth

showing, they will think worth openly receiving, and whatever it hides, worth imparting

more properly by themselves to the wider mathematical community.

PRAISE BE TO GOD

Almaty–Ankara (1998–2007)

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