**István Fenyő**'s first name appears on papers he wrote as Étienne, Stefan, Stephan or Stephen, as well as István. He was born into a family who were cultured and interested in the arts. They provided an environment in which István, from a young age, was able to enjoy a wide variety of art forms as well as interesting himself in the humanities and sciences. He made the most of these opportunities [3]:-

He studied mathematics and physics at the Pázmány Péter University in Budapest advised by Lipót Fejér who held the chair of mathematics at the University for 48 years from 1911 to 1959. In 1939 Fenyő graduated with a qualification which allowed him to teach mathematics and physics at secondary school in Hungary. However he continued his studies at the University, not in mathematics or physics but rather in chemistry. In 1942 he was awarded a Diploma in Chemistry but he undertook research in mathematics publishingEverything attracted and excited his curiosity, his insatiable thirst for knowledge and his love of life. Mathematics, technics, art, music, really every expression of human creativity, fascinated him to the extent of desiring to master whatever subject he explored.

*Über die "Polynom-Kerne" der linearen Integralgleichungen*Ⓣ in 1943. After studying for his doctorate he submitted his thesis

*On the theory of mean values*(Hungarian) in 1945. Following this he was appointed as a lecturer in the Technical University of Budapest where he was promoted to Extraordinary Professor of Mathematics in 1950, then full Professor ten years later.

The papers he published up to his promotion to extraordinary professor included *The inversion of an algorithm* (1947), (with János Aczél) *On fields of forces in which centres of gravity can be defined* (1948), (with János Aczél) *Über die Theorie der Mittelwerte* Ⓣ (1948), *Über den Mischalgorithmus der Mittelwerte* Ⓣ (1949), *The notion of mean-values of functions* (1949), and (with János Aczél and János Horváth) *Sur certaines classes de fonctionnelles* Ⓣ (1949). In addition during this period he wrote (with G Alexits) the book *Mathematics and the dialectial materialism* (Hungarian) (1948) and the paper *Les fondaments des mathématiques et la philosophie du matérialisme dialectique* Ⓣ delivered at the Tenth International Congress of Philosophy in Amsterdam in August 1949 and printed in the Proceedings in the following year.

Fenyő wrote a number of famous textbooks on applications of mathematics to chemistry and technology. His book on *Mathematics for chemists* was written jointly with G Alexits and published first in Hungarian in 1951. The third Hungarian edition of 1960 was translated into German under the title *Mathematik für Chemiker* Ⓣ (1962) and into French under the title *Les méthodes mathématiques en chimie* Ⓣ (1969). Fenyő's Hungarian text *Mathematics in electrical engineering* was written with Thomas Frey and published in two volumes, the first in 1964 with the second appearing in the following year. A Bulgarian translation of these two volumes was published in 1977 and 1979. Fenyő also wrote *Moderne mathematische Methoden in der Technik* Ⓣ, a three volume work published in 1967, 1971 and 1980. The first volume was written jointly with Thomas Frey, but the second volume was written by Fenyő alone. Both volumes were later translated into English. Keith Stewartson reviewed the first volume and we present extracts from that review:-

V Komkov reviewing the second volume of the English addition adds this interesting comment after his review:-The aim of this book is to describe some of the developments in mathematics during this century which have had or are likely to have a significant impact on the work of a theoretical engineer or technologist. The material should therefore be regarded as supplementing the methods of classical analysis, algebra and geometry which are now the bread and butter of research workers. ...

The book opens with a discussion of elementary set theory, Lebesgue integration and Stieltjes integration and then goes on to the first major topic, the operator calculus, following the ideas of Mikusinski and others. The authors prove Titchmarsh's theorem, which is central to the theory, and include a number of worked examples. The next major topic is the theory of distributions. Here, after laying the groundwork, the authors discuss sequences, Fourier transforms and the regularization of functions, and conclude with a number of applications. The last topic in the book is the theory of non-linear ordinary differential equations, beginning with questions of existence consequences and stability. The structure of the solutions is then examined, including singular points and limit cycles, and the book concludes with an account of the elementary theory of non-linear oscillations.

The third volume published in 1980, although still presenting methods for engineers, is more involved with one of Fenyő's main research topics, namely integral equations. The contents are described in a review:-Most textbooks with titles "Mathematical methods for engineers"(or for technicians, or for other users of mathematics at the elementary level)contain a mixture of topics which depend on the applications the author has in mind and on the mathematical background of the author. The unusual feature of this volume is the unified content: linear algebra, graph theory and network theory, with heavy reliance on linear algebra methods throughout.

Other books by Fenyő on integral equations areThe first section, comprising over half the book, deals with the theory of linear operators. The matters discussed are metric and normed spaces with particular reference to Hilbert spaces, Hahn-Banach theory, operators(including inverse, dual and compact operators)and eigenvalues and eigenvectors. The second section, which is over a quarter of the book, discusses linear integral equations. Amongst the topics covered are Volterra integral equations and their relation with ordinary differential equations, Fredholm equations, self-conjugate and non-self-conjugate integral operators, and the associated eigenvalue theory. The third section is on applications of integral equations. This covers topics such as the Green function technique for a one-dimensional Sturm-Liouville problem, and Dirichlet and Carl Neumann problems for the two- and three-dimensional Laplacian operator.

*Integral equations - a book of problems*(Hungarian) (1957), and the four volume work (written with H-W Stolle)

*Theorie und Praxis der linearen Integralgleichungen*Ⓣ (1982, 1983, 1983, 1984). A E Heins, reviewing the final three volumes, writes:-

In [3] Paganoni describes Fenyő's personality:-These three volumes complete the encyclopaedic work(roughly1700pages)by Fenyő and Stolle on the theory and application of linear integral equations. Their thesis is that the classical theory of linear integral equations produced many ideas for the later development of the theory of linear operators, and in turn functional analysis has helped the further development of integral equations.

He also described the range of Fenyő's interests:-An extremely cordial man, full of drive and initiative, he was a source of constant inspiration to those who had the good fortune of knowing him. He spoke several languages fluently and therefore was able to communicate directly, sharing the richness of his mind, with people of varied linguistic background. A brilliant conversationalist, with his lively anecdotal style he was able to captivate all who had the pleasure of talking to him.

... he was an extraordinarily eclectic spirit, as witnessed by the interest which, in his later years, he expressed in the mathematical aspects of Leonardo da Vinci's manuscripts. Such investigative curiosity is also clearly reflected in his mathematical studies. The soundness of his basic culture, coupled with his innate curiosity, led István Fenyő to seek the solution of problems in various areas of mathematics. Therefore it is not surprising to find, in addition to his scientific works in Mathematical Analysis, also works on the History of Mathematics, on the Philosophy of Science, and countless others on the applications of mathematics to Medicine, Engineering and Computer Science.

**Article by:** *J J O'Connor* and *E F Robertson*

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