**Samuel Verblunsky**'s parents were Elcono Verblunsky, born in Russia in 1874, and Leah Foner, born in Russia in 1880. We note that some sources give Poland as the country of birth of both Elcono Verblunsky and Leah Foner but they themselves give Russia on their census forms. A probable explanation is that since Poland did not exist as an independent country at the time of their birth, being partitioned between Russia, Prussia, and Austria, they were born in that part of Poland belonging to Russia. Elcono Verblunsky, who in the early 1900s was living at 121 Hanbury Street, Mile End New Town, London, became a naturalised British citizen on 7 August 1903. He was a tailor who had married Leah Foner in December 1901 at Whitechapel, London.

Verblunsky's school education was in London and he entered Magdalene College, Cambridge in 1924 having won a scholarship to study mathematics. Fellow mathematics students at Cambridge included Donald Coxeter, Raymond Paley, Archibald J Macintyre and James Cossar. His tutor at Magdalene College was Arthur Stanley Ramsey (1867-1954), who was president of the College and the father of the brilliant mathematical logician Frank Ramsey. Verblunsky was taught by G H Hardy and J E Littlewood and was ranked First Class in both parts of the Mathematical Tripos. After the award of his B.A. in 1927 he continued to undertake research at Magdalene College with J E Littlewood as his thesis advisor and was awarded his doctorate in 1930 for his thesis *Researches In The Theory Of Fourier Series*.

Before the award of his doctorate, Verblunsky began to publish a remarkable high volume of papers. The 1930 *Mathematical Proceedings of the Cambridge Philosophical Society* contains eight papers by Verblunsky, namely: *A property of continuous arcs *(submitted May 1929), *The relation between Riemann's method of summation and Cesàro's* (submitted May 1929),* Note on the sum of an oscillating serie*s (submitted October 1929),* Note on the Gibbs Phenomenon* (submitted November 1929), *The convergence of singular integrals *(submitted February 1930), *Note on the modified Heine-Borel theorem *(submitted April 1930),* A property of continuous arcs II *(submitted July 1930), and *Note on the sum of an oscillating serie*s *II *(submitted July 1930). These papers were all published before he was awarded a Ph.D. With eight papers appearing in these *Proceedings* in 1930 one might believe that this was the only journal in which Verblunsky published at this time. However, he was also submitting papers to the London Mathematical Society. The papers *On Summable Trigonometric Series* (submitted September 1929) and *The Generalized Third Derivative and its Application to the Theory of Trigonometric Series* (submitted September 1929) were both published in the *Proceedings* of the London Mathematical Society in 1930. His paper *On the Limit of a Function at a Point*, which he submitted to the London Mathematical Society in December 1929, appeared in the *Proceedings* in 1931. Among the results that Verblunsky proved in his early papers was that a trigonometric series cannot be summable (*C*, 1) to 0 unless its coefficients are nul, thus confirming a 1911 conjecture of Marcel Riesz.

After the award of a Ph.D. from the University of Cambridge, Verblunsky remained at Cambridge being elected as a fellow of Magdalene College. Ronald Hyam writes [5]:-

TheAt the time of his election, it emerged that he had never used a telephone and never been in a taxi, evidence of a modest life style which one older don thought 'unpropitious'.

*Mathematical Proceedings of the Cambridge Philosophical Society*of 1931 contains four Verblunsky papers, namely

*Note on Continuous Functionals*(submitted June 1930),

*The symmetric derivative and its application to the theory of trigonometric series*(submitted October 1930),

*On summable trigonometric integrals*(submitted November 1930), and

*Note on the Gibbs Phenomenon II*(submitted May 1931). At this point Verblunsky stopped publishing in the

*Mathematical Proceedings of the Cambridge Philosophical Society*and over the following few years the

*Proceedings*does not contain any of his papers. He next submitted a paper to these

*Proceedings*in May 1935, this being

*On Positive Harmonic Functions in a Half-Plane*(1935). From 1931 onwards, however, numerous papers by Verblunsky were published by the London Mathematical Society.

At some stage Verblunsky left Cambridge when he was appointed as a lecturer at the University of Manchester. We do not know the exact date on which he took up his post in Manchester but one of his 1933 papers has the University of Manchester address although some later papers still have the Magdalene College address. He remained in Manchester until 1939 and during these years he wrote two papers which are his most quoted today. These two papers, published in 1935 and 1936, were based on presentations he made to the London Mathematical Society in 1933 and 1935. The first was *On positive harmonic functions. A contribution to the algebra of Fourier series* while the second, which contains a proof of what today is called Verblunsky's Theorem, was *On positive harmonic functions*. Barry Simon writes [1]:-

Verblunsky published the textbookI got hold of[these two papers]and read them with fascination. It was hard going, but as I absorbed the papers, it became clear that there was an enormous number of ideas in these papers that had become important, but then forgotten and later rediscovered! So added to the agenda was making sure Verblunsky got the credit so long denied him!

*An introduction to the Theory of Functions of a Real Variable*in 1939. P T Daniel, reviewing this 169-page book, writes [3]:-

Another reviewer, G Baley Price, writes [2]:-This is a very good book and one which fills a long-felt need. It is scarcely necessary to say that it is a book with a conscience. The author is meticulous in all his proofs and statements and it is a pleasure to read a comparatively elementary account of the subject feeling that the author knows better than his reviewer. But there is, in addition, real artistry in designing the structure so that each theorem comes where it pulls its weight with greatest effect. The crisp style used might well be copied by other authors. Certain points are unusual and specially to be praised for neatness. ... The book is particularly suitable for the young mathematician at the University. For those at school it is too advanced, taken as a whole, except for the few. But school teachers could profitably pick out parts of it for presentation to replace proofs now fashionable. What is more important,, these teachers could be inspired by the book to treat more elementary questions; in a similar though necessarily modified way. The book would not only give aesthetic pleasure but even some instruction to University teachers.

In 1939, the year his book was published, Verblunsky was appointed as a Professor of Mathematics at Queen's University, Belfast. He remained there for the rest of his career. He joined the American Mathematical Society in 1952 at a time when he had Joseph L Smyrl as a research student. Smyrl was awarded a Ph.D. in 1953 for his thesisThis text for students and teachers was written for the special purpose of furnishing a more rigorous and accurate treatment of the elements of the theory of functions of a real variable. It is based on notes of lectures delivered by Verblunsky to students in their first year at the University of Manchester. The subject matter is entirely standard, but the treatment involves much that is new and original - ingenious and elegant proofs for certain theorems and new approaches to some parts of the subject. The dominant feature of the book is the presentation of the subject as a body of deductions from specified hypotheses. ... To this reviewer it seems that Verblunsky has succeeded admirably in his effort to write a rigorous and accurate introduction to the theory of functions of a real variable. Furthermore, the literature has been enriched permanently by original treatments of certain topics and the elegant proofs of numerous theorems which he has contributed. At the same time, the book has certain characteristics which this reviewer finds unfortunate in a text for beginners. ... The author and printer are to be congratulated on producing an excellent mathematical text.

*Some Studies in Trigonometrical Ratios*and he published the paper

*Uniqueness theorems for a class of generalized trigonometrical series*in 1954 in which he wrote:-

Another two of Verblunsky's Ph.D. students, Johnston Andrew Anderson and Gordon Harper Fullerton, wrote the joint paperI am indebted to Professor S Verblunsky, under whose supervision this work was carried out, for his helpful criticism and advice.

*On a class of Cauchy exponential series*(1965) in which they write:-

After he retired from his chair at Queen's University, Belfast, Verblunsky moved away from the city but remained living in Northern Ireland. He never married and when he died in 1996 at the age of 90 he left a bequest to the London Mathematical Society to create a reading room in his name. The Verblunsky Room is in De Morgan House, Russell Square, London and at present it is a Members' Room housing the London Mathematical Society's special collections consisting of the Hardy Collection and the Philippa Fawcett Collection.... the authors wish to express their gratitude to Professor S Verblunsky of Belfast, for his helpful criticism and advice.

Finally, let us note that MathSciNet lists 17 papers with 'Verblunsky coefficients' in the title and another 73 with this term in the review text. All of these papers have been published since the publication of Barry Simon's book [1] in 2004. It also lists 10 papers with 'Verblunsky parameters' in the review text, all published since 2007.

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