Ralph Henstock (2 June 1923 – 17 January 2007) was an English mathematician and author. As an Integration theorist, he is notable for Henstock–Kurzweil integral. Henstock brought the theory to a highly developed stage without ever having encountered Jaroslav Kurzweil's 1957 paper on the subject.

Early life

Henstock was born in the coal-mining village of Newstead, Nottinghamshire, the only child of mineworker and former coalminer William Henstock and Mary Ellen Henstock (née Bancroft). On the Henstock side he was descended from 17th century Flemish immigrants called Hemstok. Because of his early academic promise it was expected that Henstock would attend the

University of Nottingham The University of Nottingham is a public research university in Nottingham, England. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. Nottingham's main campus (University Park Campus, Nottingh ...

where his father and uncle had received technical education, but as it turned out he won scholarships which enabled him to study mathematics at

St John's College, Cambridge St John's College, formally the College of St John the Evangelist in the University of Cambridge, is a Colleges of the University of Cambridge, constituent college of the University of Cambridge, founded by the House of Tudor, Tudor matriarch L ...

from October 1941 until November 1943, when he was sent for war service to the

Ministry of Supply The Ministry of Supply (MoS) was a department of the UK government formed on 1 August 1939 by the Ministry of Supply Act 1939 ( 2 & 3 Geo. 6. c. 38) to co-ordinate the supply of equipment to all three British armed forces, headed by the Ministe ...

's department of Statistical Method and Quality Control in London. This work did not satisfy him, so he enrolled at

Birkbeck College Birkbeck, University of London (formally Birkbeck College, University of London), is a public research university located in London, England, and a member institution of the University of London. Established in 1823 as the London Mechanics' ...

, London where he joined the weekly seminar of Professor

Paul Dienes Paul Dienes (Hungarian people, Hungarian: ''Dienes Pál''. November 24, 1882 Tokaj, Austria-Hungary – March 23, 1952) was a Hungarian mathematician, philosopher, linguist and poet. Born in to a wealthy and aristocratic Protestant family, he mar ...

which was then a focus for mathematical activity in London. Henstock wanted to study

divergent series In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. If a series converges, the individual terms of the series mus ...

but Dienes prevailed upon him to get involved in the theory of integration, thereby setting him on course for his life's work. A devoted

Methodist Methodism, also called the Methodist movement, is a Protestant Christianity, Christian Christian tradition, tradition whose origins, doctrine and practice derive from the life and teachings of John Wesley. George Whitefield and John's brother ...

, the lasting impression he made was one of gentle sincerity and amiability. Henstock married Marjorie Jardine in 1949. Their son John was born 10 July 1952. Ralph Henstock died on 17 January 2007 after a short illness.

Work

Henstock was awarded the Cambridge B.A. in 1944 and began research for the

PhD A Doctor of Philosophy (PhD, DPhil; or ) is a terminal degree that usually denotes the highest level of academic achievement in a given discipline and is awarded following a course of graduate study and original research. The name of the deg ...

in Birkbeck College, London, under the supervision of Paul Dienes. His PhD thesis, entitled
Interval Functions and their Integrals
', was submitted in December 1948. His Ph.D. examiners were Burkill and H. Kestelman. In 1947 he returned briefly to Cambridge to complete the undergraduate mathematical studies which had been truncated by his Ministry of Supply work. Most of Henstock's work was concerned with integration. From initial studies of the Burkill and

Ward Ward may refer to: Division or unit * Hospital ward, a hospital division, floor, or room set aside for a particular class or group of patients, for example the psychiatric ward * Prison ward, a division of a penal institution such as a pris ...

integrals he formulated an integration process whereby the domain of integration is suitably partitioned for

Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approxima ...

s to approximate the integral of a function. His methods led to an integral on the real line that was very similar in construction and simplicity to the

Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...

but which included the

Lebesgue integral In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

and, in addition, allowed non-absolute convergence. These ideas were developed from the late 1950s. Independently, Jaroslav Kurzweil developed a similar Riemann-type integral on the real line. The resulting integral is now known as the Henstock-Kurzweil integral. On the real line it is equivalent to the Denjoy-Perron integral, but has a simpler definition. In the following decades, Henstock developed extensively the distinctive features of his theory, inventing the concepts of division spaces or integration bases to demonstrate in general settings the properties and characteristics of mathematical integration. His theory provides a unified approach to non-absolute integral, as different kinds of Henstock integral, choosing an appropriate integration basis (division space, in Henstock's own terminology). It has been used in differential and integral equations, harmonic analysis, probability theory and Feynman integration. Numerous monographs and texts have appeared since 1980 and there have been several conferences devoted to the theory. It has been taught in standard courses in mathematical analysis. Henstock was author of 46 journal papers in the period 1946 to 2006. He published four books on analysis (Theory of Integration, 1963; Linear Analysis, 1967; Lectures on the Theory of Integration, 1988; and The General Theory of Integration, 1991). He wrote 171 reviews for

MathSciNet MathSciNet is a searchable online bibliographic database created by the American Mathematical Society in 1996. It contains all of the contents of the journal ''Mathematical Reviews'' (MR) since 1940 along with an extensive author database, links ...

. In 1994 he was awarded the Andy Prize of the XVIII Summer

Symposium In Ancient Greece, the symposium (, ''sympósion'', from συμπίνειν, ''sympínein'', 'to drink together') was the part of a banquet that took place after the meal, when drinking for pleasure was accompanied by music, dancing, recitals, o ...

in Real Analysis. His academic career began as Assistant Lecturer, Bedford College for Women, 1947–48; then Assistant Lecturer at Birkbeck, 1948–51; Lecturer,

Queen's University Belfast The Queen's University of Belfast, commonly known as Queen's University Belfast (; abbreviated Queen's or QUB), is a public research university in Belfast, Northern Ireland, United Kingdom. The university received its charter in 1845 as part of ...

, 1951–56; Lecturer,

Bristol University The University of Bristol is a public research university in Bristol, England. It received its royal charter in 1909, although it can trace its roots to a Merchant Venturers' school founded in 1595 and University College, Bristol, which had ...

, 1956–60; Senior Lecturer and Reader, Queen's University Belfast, 1960–64; Reader,

Lancaster University Lancaster University (officially The University of Lancaster) is a collegiate public university, public research university in Lancaster, Lancashire, England. The university was established in 1964 by royal charter, as one of several new univer ...

, 1964–70; Chair of Pure Mathematics,

New University of Ulster Ulster University (; Ulster Scots dialects, Ulster Scots: or ), legally the University of Ulster, is a multi-campus Public university, public research university located in Northern Ireland. It is often referred to informally and unofficially a ...

, 1970–88; and Leverhulme Fellow 1988–91.

List of publications of Ralph Henstock

Much of Henstock's earliest work was published by the

Journal of the London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

. These were "On interval functions and their integrals" I (21, 1946) and II (23, 1948); "The efficiency of matrices for

Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

" (22, 1947); "The efficiency of matrices for bounded sequences" (25, 1950); "The efficiency of convergence factors for functions of a continuous real variable" (30, 1955); "A new description of the Ward integral" (35 1960); and "The integrability of functions of interval functions" (39 1964). His works, published in ''Proceedings of the

London Mathematical Society The London Mathematical Society (LMS) is one of the United Kingdom's Learned society, learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh ...

'', were "Density integration" (53, 1951); "On the measure of sum sets (I) The theorems of Brunn, Minkowski, and Lusternik, (with A.M. McBeath)" ( 3, 1953); "Linear functions with domain a real countably infinite dimensional space" ( 5, 1955); "Linear and bilinear functions with domain contained in a real countably infinite dimensional space" ( 6, 1956); "The use of convergence factors in Ward integration" ( 10, 1960); "The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals" ( 10, 1960); "N-variation and N-variational integrals of set functions" ( 11, 1961); "Definitions of Riemann type of the variational integrals" ( 11, 1961); "Difference-sets and the

Banach–Steinhaus theorem In mathematics, the uniform boundedness principle or Banach–Steinhaus theorem is one of the fundamental results in functional analysis. Together with the Hahn–Banach theorem and the open mapping theorem, it is considered one of the cornersto ...

" ( 13, 1963); "Generalized integrals of vector-valued functions ( 19 1969) Additional publications: #Sets of uniqueness for trigonometric series and integrals, Proceedings of the

Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ...

46 (1950) 538–548. #On Ward's Perron-Stieltjes integral, Canadian Journal of Mathematics 9 (1957) 96–109. #The summation by convergence factors of Laplace-Stieltjes integrals outside their half plane of convergence, Mathematische Zeitschrift 67 (1957) 10–31. #Theory of Integration, Butterworths, London, 1962.

' #

Tauberian theorems In mathematics, Abelian and Tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber. The original examples are Abel's theorem showing th ...

for integrals, Canadian Journal of Mathematics 15 (1963) 433–439. #Majorants in variational integration, Canadian Journal of Mathematics 18 (1966) 49–74. #A Riemann-type integral of Lebesgue power, Canadian Journal of Mathematics 20 (1968) 79–87. #Linear Analysis, Butterworths, London, 1967.

' #Integration by parts, Aequationes Mathematicae 9 (1973) 1–18. #The N-variational integral and the Schwartz distributions III, Journal of the London Mathematical Society (2) 6 (1973) 693–700. #Integration in product spaces, including Wiener and Feynman integration, Proceedings of the London Mathematical Society (3) 27 (1973) 317–344. #Additivity and the Lebesgue limit theorems, The Greek Mathematical Society C. Carathéodory Symposium, 1973, 223–241 (Proceedings published 1974).

' #Integration, variation and differentiation in division spaces, Proceedings of the Royal Irish Academy, Series A (10) 78 (1978) 69–85. #The variation on the real line, Proceedings of the Royal Irish Academy, Series A (1) 79 (1979) 1–10. #Generalized Riemann integration and an intrinsic topology, Canadian Journal of Mathematics 32 (1980) 395–413. #Division spaces, vector-valued functions and backwards martingales, Proceedings of the Royal Irish Academy, Series A (2) 80 (1980) 217–232. #Density integration and Walsh functions, Bulletin of the Malaysian Mathematical Society (2) 5 (1982) 1–19. #A problem in two-dimensional integration, Journal of the Australian Mathematical Society, (Series A) 35 (1983) 386–404. #The Lebesgue syndrome, Real Analysis Exchange 9 (1983–84) 96–110. #The reversal of power and integration, Bulletin of the Institute of Mathematics and its Applications 22 (1986) 60–61. #Lectures on the Theory of Integration, World Scientific, Singapore, 1988.

' #A short history of integration theory, South East Asian Bulletin of Mathematics 12 (1988) 75–95. #Introduction to the new integrals, New integrals (Coleraine, 1988), 7–9, Lecture Notes in Mathematics, 1419, Springer-Verlag, Berlin, 1990. #Integration in infinite-dimensional spaces, New integrals (Coleraine, 1988), 54–65, Lecture Notes in Mathematics, 1419, Springer-Verlag, Berlin, 1990. #Stochastic and other functional integrals, Real Analysis Exchange 16 (1990/91) 460–470. #The General Theory of Integration, Oxford Mathematical Monographs, Clarendon Press, Oxford, 1991.

' #The integral over product spaces and Wiener's formula, Real Analysis Exchange 17 (1991/92) 737–744. #Infinite decimals, Mathematica Japonica 38 (1993) 203–209. #Measure spaces and division spaces, Real Analysis Exchange 19 (1993/94) 121–128. #The construction of path integrals, Mathematica Japonica 39 (1994) 15–18. #Gauge or Kurzweil-Henstock integration. Proceedings of the Prague Mathematical Conference 1996, 117–122, Icaris, Prague, 1997. #De La Vallée Poussin's contributions to integration theory, Charles-Jean de La Vallée Poussin Oeuvres Scientifiques, Volume II, Académie Royale de Belgique, Circolo Matematico di Palermo, 2001, 3–16. #Partitioning infinite-dimensional spaces for generalized Riemann integration, (with P. Muldowney and V.A. Skvortsov) Bulletin of the London Mathematical Society, 38 (2006) 795–803.

Review of Henstock's work

The journal Scientiae Mathematicae Japonicae published a special commemorative issue in Henstock’s honor, January 2008. The above article is copied, with permission, from Real Analysis Exchange and from Scientiae Mathematicae Japonicae. The latter contains the following review of Henstock's work: 1. Ralph Henstock, an obituary, by P. Bullen. 2. Ralph Henstock: research summary, by E. Talvila. 3. The integral à la Henstock, by Peng Yee Lee. 4. The natural integral on the real line, by B. Thomson. 5. Ralph Henstock's influence on integration theory, by W.F. Pfeffer. 6. Henstock on random variation, by P. Muldowney. 7. Henstock integral in harmonic analysis, by V.A. Skvortsov. 8. Convergences on the Henstock-Kurzweil integral, by S. Nakanishi.

External links

The Calculus and Gauge Integrals
by Ralph Henstock
Lectures on Integration
by Ralph Henstock
Autobiographical notes
by Ralph Henstock

References