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This text for undergraduate and graduate students illustrates the fundamental simplicity of the properties of orthogonal functions and their developments in related series. Starting with a definition and explanation of the elements of Fourier series, the text follows with examinations of Legendre polynomials and Bessel functions. Boundary value problems consider Fourier series in conjunction with Laplace's equation in an infinite strip and in a rectangle, with a vibrating string, in three dimensions, in a sphere, and in other circumstances. An overview of Pearson frequency functions is followed by chapters on orthogonal, Jacobi, Hermite, and Laguerre polynomials, and the text concludes with a chapter on convergence. 1941 edition.
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Jackson was born on July 24, 1888, in Bridgewater, Massachusetts. His parents, William Dunham Jackson and Mary Vose (Morse) Jackson, were graduates of the Normal School at Bridgewater, and both were descended from passengers on the Mayflower. William Jackson became a science and mathematics professor at the Bridgewater Normal School and was active in the Congregationalist church.
At an early age Dunham Jackson joined his father for walks to study geology, botany, and zoology; he was soon reading science textbooks in the family library. At the local high school his favorite subjects were languages and literature, and he enjoyed teaching Latin and German poetry to his younger sister, Elizabeth. Jackson entered Harvard University when he was sixteen. After receiving the B. A. degree in 1908, he held an assistantship in astronomy until he received his M. A. in 1909.
His many honors included membership in Phi Beta Kappa and several fellowships to support doctoral studies abroad. In Germany, Jackson studied at the University of Göttingen in 1909-1911 under the tutelage of Edmund Landau and spent a few months at the University of Bonn in 1911. He was awarded the Ph. D. and a prize at Göttingen in 1911.
Jackson became an instructor in mathematics at Harvard in 1911 and an assistant professor in 1916. While Jackson served as a captain in the Ordnance Department of the U. S. Army in Washington, D. C. , from November, 1918, to August, 1919, he wrote a pamphlet-text on numerical integration in exterior ballistics for the department.
In 1919 Jackson accepted a professorship of mathematics at the University of Minnesota, where he remained until his death. There, his expository writing and speaking, his persistence in continuing undergraduate teaching, his stimulus of graduate students to scholarship in his special area, and his extended participation in editorial and organizational activities all significantly extended his influence. Jackson's greatest contribution, however, was probably his own research.
At least seventy-five of his works involved significant novelty in content, methods, or organization. Jackson's first mathematical research paper, published in 1909, was algebraic, dealing with transformations of bilinear forms. Most of his papers after his return from Göttingen were within the broad field related to his dissertation. He wrote extensively on orthogonal polynomials, trigonometric sums, and their relation to the theory of approximations, all important fields today. However, he showed substantial diversity in his explorations of the connections of these topics.
In 1930 the American Mathematical Society published Jackson's The Theory of Approximation, the outgrowth of the Colloquium Lectures he gave at the society's summer meeting in 1925. Jackson was also invited to lecture to the society and its Chicago Section in 1921, 1928, 1933, and 1934. In 1933 he was awarded the Mathematical Association of America's Chauvenet prize for the best expository writing over the previous three years. In 1944 the Association published his Fourier Series and Orthogonal Polynomials as the sixth of its Carus Mathematical Monographs. His many expository articles and notes included seventeen in the American Mathematical Monthly.
Jackson's concerns for teaching and exposition were also reflected in his serving in 1929-1931 as chairman of the joint Committee on Geometry of the Mathematical Association and the National Council of Teachers of Mathematics, and his work at different times as editor, associate, and assistant editor of the Bulletin of the American Mathematical Society (1921-1925) and the Transactions of the American Mathematical Society (1916-1925, 1926-1931). Jackson never regained full health after a heart attack in 1940. The last eighteen months of his life were spent at home and in the hospital. However, even during this time he approved at least four theses and completed a research paper. He died in Minneapolis on November 6, 1946, and is buried there in Sunset Memorial Park.
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Jackson was a member of Sigma Xi, a member of the council of the American Mathematical Society (1918-1920). Other posts included the President of the Mathematical Association of America (1926), vice-president of the American Association for the Advancement of Science (1927), member of the Institute of Mathematical Statistics, fellow of the American Academy of Arts and Sciences, fellow of the American Physical Society, and member of the National Academy of Science (1935).
On June 20, 1918, Jackson married Harriet Spratt Hulley, whom he had met while she was a graduate student in English at Radcliffe College. They had two daughters, Anne Hulley and Mary Eloise.
