**Dunham Jackson**was the son of William Dunham Jackson and Mary Vose. William Jackson was a teacher of science and mathematics at the Normal School in Bridgewater, the school that both he and Mary Vose had attended. Dunham was an exceptional child, learning a broad range of subjects from his father [5]:-

Dunham attended both elementary school and high school in Bridgewater. He excelled in high school, not only in science subjects but also in English literature, Latin, Greek and German. He loved poetry and learnt a wide range of English poetry by heart and, perhaps even more remarkably, also poetry in Latin and German. He delighted in teaching this poetry to his younger sister while they did household chores together. In addition to these academic interests, he loved many sports, enjoying baseball, tennis, skating, cycling, swimming, sailing, and hiking. In 1904, when only sixteen years old, Jackson began his studies at Harvard University.As soon as Jackson was of walking age, he became his father's companion on long hikes and his father's interest in science made these excursions essentially field trips for the study of botany, zoology, and geology. Thus, very early, Jackson acquired from his father a vast store of miscellaneous information and habits of observation that he never outgrew. At an early stage, he started reading the science textbooks in his father's library.

Although mathematics was the main subject that Jackson studied at Harvard, he also took courses in astronomy, chemistry, physics, classical languages and modern languages. His work was outstanding and in recognition of his achievements he was awarded a Wendell Scholarship, and later a Palfrey Exhibition and the Richard Augustine Gambrill Scholarship. During these undergraduate years the lecturer who influenced him most was Maxime Bôcher; it was advice from Bôcher that led to Jackson's first paper *Resolution into Involutory Substitutions of the Transformations of a Nonsingular Bilinear Form into Itself* (1909) which he wrote while still an undergraduate. Jackson graduated from Harvard with an A.B (with distinction in mathematics) in 1908 and an A.M. in 1909. During 1908-09 while he was studying for his Master's degree he held a George C Shattuck Fellowship and was appointed as an assistant in astronomy. Harvard awarded him the Rogers Fellowship for his studies at Göttingen for 1909-10 and the Edward William Hopper Fellowship for the following year.

At Göttingen, Jackson learnt much from David Hilbert, Felix Klein and Ernst Zermelo, but the strongest influence on him was Edmund Landau who was his advisor. During these two years abroad he also visited Bonn, where he spent two months attending lectures by Felix Hausdorff and Eduard Study, and Paris where he spent a few weeks attending lectures by Émile Picard, Édouard Goursat and Jacques Hadamard. Jackson was awarded his doctorate by Göttingen in 1911 for his dissertation *Über die Genauigkeit der Annäherung stetiger Funktionen durch ganze rationale Funktionen gegebenen Grades und trigonometrische Summen gegebener Ordnung* Ⓣ. Jackson wrote in the Preface to *The theory of approximation* (1930):-

We should note that in fact Jackson extended the topic from that proposed by Edmund Landau to include approximation by trigonometric sums in addition to approximation by polynomials. William Hart writes [5]:-Guided partly by natural inclinations, perhaps, and partly by recollection of a course on methods of approximation which I had taken with Professor Bôcher a few years earlier, I committed myself to one of the topics which Edmund Landau had proposed for a thesis, an investigation of the degree of approximation with which a given continuous function can be represented by a polynomial of given degree. When I reported my choice he said meditatively "... Das ist ein schönes Thema, ich beneide Sie um das Thema ... Nein, ich beneide Sie nicht, aber es ist ein wunderschönes Thema."Ⓣ

We should note that Jackson produced this remarkable thesis despite having health problems. In the spring of 1910 he contracted polio but somehow managed to continue his research throughout the illness which left him with a permanent limp. He kept news of the illness from his family and they only discovered what he had gone through after he returned to his home town after defending his thesis on 12 July 1911. Given the outstanding achievements in the thesis it will come as no surprise to learn that he received the degree with distinction.Jackson's thesis answered a prize question which had been proposed earlier by the Göttingen faculty: "Would it be possible to improve on results(on approximation by polynomials)previously obtained by de la Vallee-Poussin and Lebesgue?" In a complimentary statement awarding Jackson the prize, we find the remark: "The author ... has enriched the science with valuable results, all in competition with mathematicians of the first rank."

It is no reflection on Jackson's later achievements to remark that his thesis was one of his most important pieces of mathematical research. The fact that his results improved on those of very famous mathematicians gives sufficient assurance of the quality of his thesis, and Landau was amply justified when he labelled its topic "ein wunderschönes Thema." Jackson thus had the rare good fortune to produce, in his first major effort, results which were fundamental for the development of a major field.

Back in the United States, Jackson was appointed as an Instructor in Mathematics at Harvard in 1911 and, five years later, was promoted to Assistant Professor. In 1917 he met Harriet Spratt Hulley who at that time was undertaking graduate studies in English at Radcliffe College. They were married on 20 June 1918; Dunham and Harriet Jackson had two daughters, Anne Hulley Jackson and Mary Eloise Jackson [4]:-

By the time he was married, Jackson was involved in war work becoming commissioned as an officer in the Ordnance Department. He was sent to work at the Ballistic Unit of the Ordnance Department in Washington, D. C., undertaking the calculation of range tables for artillery. While working there he prepared a 43-page instruction bookletAs an outstanding characteristic, he enjoyed his associations with people. In particular, his congenial wife and family were a continuing source of pleasure and inspiration to him.

*The Method of Numerical Integration in Exterior Ballistics*. He resigned his commission in the spring of 1919 when offered a Professorship in Mathematics at the University of Minnesota. In fact the offer from Minnesota was a difficult one for Jackson for, on the one hand, it was a major promotion but, on the other hand, he was moving from Harvard with its big graduate school in mathematics to a place with no graduate school in the subject. However, he was attracted by the opportunity that Minnesota would give him to develop a graduate school and he took up the chair in September 1919. He continued to hold this chair until his death some 27 years later preferring to remain at Minnesota despite receiving a number of attractive offers of chairs in other universities.

Jackson published many important papers and rapidly gained an outstanding reputation as an excellent speaker. It was natural, therefore that the American Mathematical Society would invite him to be the principal speaker at its 1921 Symposium: he took as his title *The General Theory of Approximation by Polynomials and Trigonometric Sums*. Four years later the Society honoured him with an invitation to be their Colloquium Lecturer. Edward Copson writes:-

The lectures were published asIn1925, Professor Jackson lectured at the American Mathematical Society's Colloquium on certain aspects of the theory of approximation, i.e. on the theory of approximating to a function of a real variable by means of a finite sum of functions of given form. This theory covers a vast field, and, in his lectures, Professor Jackson chose certain topics in a few corners of this field in which he is personally interested and where he has made valuable contributions.

*The Theory of Approximation*(1930) and, in the Preface, Jackson explains the selection of topics:-

J Shohat reviews the text giving it high praise [9]:-The title of this volume is an abbreviation for the more properly descriptive one: 'Topics in the Theory of Approximation'. It is an account of certain aspects and ramifications of a problem to which I was introduced at an early stage, and which has given direction to my reading and study ever since.

Perhaps the lasting importance of this book is best illustrated by noting that the American Mathematical Society reprinted the 1930 original in 1994. As well as this monograph, Jackson published a student textThe reader will enjoy it as interesting, stimulating, extremely well written, and beautifully printed. Many a worker will be inspired by the pages devoted to Fourier and Legendre series and to Chebyshev-Hermite polynomials to try his hand in obtaining similar results for other classes of orthogonal functions, and in particular, for other classes of orthogonal Chebyshev polynomials.

*Fourier Series and Orthogonal Polynomials*in 1941 in the Carus Mathematical Monographs of the Mathematical Association of America. He explains his position on mathematical rigour in the Preface:-

Copson writes [2]:-Under the circumstances, 'rigour' in the sense of literal completeness of statement has been out of the question. It is hoped, however, that the reader who is familiar with the methods of rigorous analysis will be able without any difficulty to read between the lines the requisite supplementary specifications, and will find that what has actually been said is entirely accurate in the light of such interpretation.

Shohat [10] begins his review by explaining that the book:-The task which the author has set himself is no easy one, but he has succeeded in producing a sound and very readable book well suited to the purpose he had in view.

Curtiss writes [3]:-... measures up fully to the expectation of the reader who is acquainted(and who is not?)with the lucid and elegant presentation - oral and written - of Dunham Jackson.

Charles Moore write in his review [6]:-The exposition is well organized; much of the formal work is beautifully handled; the proofs are extremely clear. It will not be surprising if this turns out to be one of the most popular of the Carus Monographs. The Association is indeed fortunate to be able to add to the Carus series an exposition by one of the world's foremost authorities on orthogonal polynomials.

In 1940 Jackson suffered a major heart attack. After recovering he had two years during which he was able to live a reasonably normal life although he had to take things easily. Then he began to suffer a series of minor heart attacks which prevented him from carrying out his teaching duties. By 1943 his life consisted either of lying in a hospital bed or in a bed in his own home. Despite these severe health problems he continued to supervise doctoral students and to write research papers.A proper balance between formal applications and rigorous analysis has been maintained, and the approach in general has the clarity and elegance that one has come to expect from this particular author.

Let us end this biography by looking briefly at his character and his views on certain matters. As to his character, Hart writes [4]:-

The article [5] contains some fascinating extracts from a diary that Jackson wrote while undertaking research in Germany. We give, as an example, just one of these extracts where Jackson is musing about teaching mathematics at an advanced level:-Jackson was a man of high ideals, extremely unselfish, and very conscious of his responsibilities, not only in strictly personal affairs but also with respect to his profession as an educator and his duties as a citizen. He was intensely devoted to the field of mathematics, had thought deeply of its place in advancing human welfare, and was fluent in expressing his convictions on such matters.

What is the purpose of a higher course in mathematics? To teach facts or methods? Should a course be simply a succession of theorems with proofs? Or, is it undesirable to try to prove everything? Does a mathematician read and make absolutely clear to himself all the work on which his own original work is based, so that he can assert, of his own knowledge, that the preceding work is true? Or, does he leave something to the accuracy of his predecessor? This is a point on which information ought to be gained by inquiry. Is the purpose of proofs in a higher course to teach methods which will be useful to the pupil or to assure him that every point is based on a rigorous foundation? If the answer to these questions is different for different courses or for different parts of a course, I think it would be well for the teacher to have in mind just what point of view he is adopting at each stage.

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