Ralph Henstock
Encyclopedia
Ralph Henstock 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.
, 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 Nottingham University 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
from October 1941 until November 1943, when he was sent for war service to the Ministry of Supply
’s department of Statistical Method and Quality Control in London.
This work did not satisfy him, so he enrolled at Birkbeck College
, London
where he joined the weekly seminar of Professor Paul Dienes which was then a focus for mathematical activity in London. Henstock wanted to study divergent series
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, the lasting impression he made was one of gentle sincerity and amiability. Henstock married Marjorie Jardine in 1949. Their son John was born July 10 1952. Ralph Henstock died on January 17 2007 after a short illness.
Most of Henstock's work was concerned with integration. From initial studies of the Burkill and Ward
integrals he formulated an integration process whereby the domain of integration is suitably partitioned for Riemann sum
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
but which included 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. In 1994 he was awarded the Andy Prize of the XVIII Summer Symposium
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, 1951-56; Lecturer, Bristol University, 1956-60; Senior Lecturer and Reader, Queen’s University Belfast, 1960-64; Reader, Lancaster University
, 1964-70; Chair of Pure Mathematics, New University of Ulster, 1970-88; and Leverhulme Fellow 1988-91.
" (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
, were "Density integration" (53, 1951); "On the measure of sum sets (I) The theorems of Brunn, Minkowski, and Lusternik, (with A.M. McBeath)" ([3] 3, 1953); "Linear functions with domain a real countably infinite dimensional space" ([3] 5, 1955); "Linear and bilinear functions with domain contained in a real countably infinite dimensional space" ([3] 6, 1956); "The use of convergence factors in Ward integration" ([3] 10, 1960); "The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals" ([3] 10, 1960); "N-variation and N-variational integrals of set functions" ([3] 11, 1961); "Definitions of Riemann type of the variational integrals" ([3] 11, 1961); "Difference-sets and the Banach–Steinhaus theorem" ([3] 13, 1963); "Generalized integrals of vector-valued functions ([3] 19 1969)
His additional papers were
1. Ralph Henstock, an obituary, by P. Bullen.
2. Ralph Henstock: research summary, by E. Talvila.
3. The integral à la Henstock, by Lee P.Y.
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.
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
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
Jaroslav Kurzweil
Jaroslav Kurzweil is a Czech mathematician. He is a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riemann sums...
's 1957 paper on the subject.
Early life
He was born in the coal-mining village of Newstead, NottinghamshireNewstead, Nottinghamshire
Newstead is a small village and civil parish in Nottinghamshire, England, situated between Nottingham and Mansfield in the Borough of Gedling. It is a former coal mining village, and was previously called Newstead Colliery Village. Lord Byron, the poet, lived at nearby Newstead Abbey. The parish...
, 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
Flemish people
The Flemings or Flemish are the Dutch-speaking inhabitants of Belgium, where they are mostly found in the northern region of Flanders. They are one of two principal cultural-linguistic groups in Belgium, the other being the French-speaking Walloons...
immigrants called Hemstok.
Because of his early academic promise it was expected that Henstock would attend Nottingham University 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, Cambridge
St John's College is a constituent college of the University of Cambridge. The college's alumni include nine Nobel Prize winners, six Prime Ministers, three archbishops, at least two princes, and three Saints....
from October 1941 until November 1943, when he was sent for war service to the Ministry of Supply
Ministry of Supply
The Ministry of Supply was a department of the UK Government formed in 1939 to co-ordinate the supply of equipment to all three British armed forces, headed by the Minister of Supply. There was, however, a separate ministry responsible for aircraft production and the Admiralty retained...
’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
Birkbeck, University of London is a public research university located in London, United Kingdom and a constituent college of the federal University of London. It offers many Master's and Bachelor's degree programmes that can be studied either part-time or full-time, though nearly all teaching is...
, London
London
London is the capital city of :England and the :United Kingdom, the largest metropolitan area in the United Kingdom, and the largest urban zone in the European Union by most measures. Located on the River Thames, London has been a major settlement for two millennia, its history going back to its...
where he joined the weekly seminar of Professor Paul Dienes which was then a focus for mathematical activity in London. Henstock wanted to study divergent series
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 limit....
but Dienes prevailed upon him to get involved in the theory of integration
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
, thereby setting him on course for his life’s work.
A devoted Methodist, the lasting impression he made was one of gentle sincerity and amiability. Henstock married Marjorie Jardine in 1949. Their son John was born July 10 1952. Ralph Henstock died on January 17 2007 after a short illness.
Work
He was awarded the Cambridge B.A. in 1944 and began research for the PhD in London, which he gained in December 1948 with a thesis entitled Interval Functions and their Integrals, an extension of J. C. Burkill's theory. 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
John Clive Ward
John Clive Ward , was a British-Australian physicist. His most famous creation was the Ward-Takahashi identity, originally known as "Ward Identity" . This celebrated result, in quantum electrodynamics, was inspired by a conjecture of Dyson and was disclosed in a one-half page letter typical of...
integrals he formulated an integration process whereby the domain of integration is suitably partitioned for Riemann sum
Riemann sum
In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....
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
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. The Riemann integral is unsuitable for many theoretical purposes...
but which included 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
Henstock-Kurzweil integral
In mathematics, the Henstock–Kurzweil integral, also known as the Denjoy integral and the Perron integral, is one of a number of definitions of the integral of a function. It is a generalization of the Riemann integral which in some situations is more useful than the Lebesgue integral.This 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. In 1994 he was awarded the Andy Prize of the XVIII Summer Symposium
Academic conference
An academic conference or symposium is a conference for researchers to present and discuss their work. Together with academic or scientific journals, conferences provide an important channel for exchange of information between researchers.-Overview:Conferences are usually composed of various...
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, 1951-56; Lecturer, Bristol University, 1956-60; Senior Lecturer and Reader, Queen’s University Belfast, 1960-64; Reader, Lancaster University
Lancaster University
Lancaster University, officially The University of Lancaster, is a leading research-intensive British university in Lancaster, Lancashire, England. The university was established by Royal Charter in 1964 and initially based in St Leonard's Gate until moving to a purpose-built 300 acre campus at...
, 1964-70; Chair of Pure Mathematics, New University of Ulster, 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. These were "On interval functions and their integrals" I (21, 1946) and II (23, 1948); "The efficiency of matrices for Taylor seriesTaylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
" (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
London Mathematical Society
-See also:* American Mathematical Society* Edinburgh Mathematical Society* European Mathematical Society* List of Mathematical Societies* Council for the Mathematical Sciences* BCS-FACS Specialist Group-External links:* * *...
, were "Density integration" (53, 1951); "On the measure of sum sets (I) The theorems of Brunn, Minkowski, and Lusternik, (with A.M. McBeath)" ([3] 3, 1953); "Linear functions with domain a real countably infinite dimensional space" ([3] 5, 1955); "Linear and bilinear functions with domain contained in a real countably infinite dimensional space" ([3] 6, 1956); "The use of convergence factors in Ward integration" ([3] 10, 1960); "The equivalence of generalized forms of the Ward, variational, Denjoy-Stieltjes, and Perron-Stieltjes integrals" ([3] 10, 1960); "N-variation and N-variational integrals of set functions" ([3] 11, 1961); "Definitions of Riemann type of the variational integrals" ([3] 11, 1961); "Difference-sets and the Banach–Steinhaus theorem" ([3] 13, 1963); "Generalized integrals of vector-valued functions ([3] 19 1969)
His additional papers were
- Sets of uniqueness for trigonometric series and integrals, Proceedings of the Cambridge Philosophical SocietyCambridge Philosophical SocietyThe Cambridge Philosophical Society is a scientific society at 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 theology and medicine...
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 integralsLaplace-Stieltjes transformThe Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform. For real-valued functions, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a Banach...
outside their half plane of convergence, Mathematische Zeitschrift 67 (1957) 10-31. - Theory of Integration, Butterworths, London, 1962.
- Tauberian theorems 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, 1968.
- Integration by parts, Aequationes Mathematicae 9 (1973) 1-18.
- The N-variational integral and the Schwarz 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. Caratheodory Symposium, 1973, 223-241, (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 his 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 Lee P.Y.
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.