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 In mathematics, the Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced ), Luzin integral or Perron integral, but not to be confused with the more general wide ...

. Henstock brought the theory to a highly developed stage without ever having encountered

Jaroslav Kurzweil Jaroslav Kurzweil (, 7 May 1926, Prague – 17 March 2022) was a Czech mathematician. Biography Born in Prague, Czechoslovakia, he was a specialist in ordinary differential equations and defined the Henstock–Kurzweil integral in terms of Riema ...

's 1957 paper on the subject.

Early life

He was born in the coal-mining village of

Newstead, Nottinghamshire Newstead is a village and civil parish in Nottinghamshire, England in the borough of Gedling.OS Explorer Map 270: Sherwood Forest: (1:25 000): It is situated between the city of Nottingham and the towns of Kirkby-in-Ashfield, Sutton-in-Ashfield ...

, 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 (''Vlaams'') is a Low Franconian dialect cluster of the Dutch language. It is sometimes referred to as Flemish Dutch (), Belgian Dutch ( ), or Southern Dutch (). Flemish is native to Flanders, a historical region in northern Belgium; ...

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 university, public research university in Nottingham, United Kingdom. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. The University of Nottingham belongs t ...

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 is a Colleges of the University of Cambridge, constituent college of the University of Cambridge founded by the House of Tudor, Tudor matriarch Lady Margaret Beaufort. In constitutional terms, the college is a charitable corpo ...

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 in 1939 to co-ordinate the supply of equipment to all three British armed forces, headed by the Minister of Supply. A separate ministry, however, was responsible for aircr ...

'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 university, public research university, located in Bloomsbury, London, England, and a constituent college, member institution of the federal Universit ...

, London where he joined the weekly seminar of Professor

Paul Dienes Paul Dienes ( 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 married Valéria Ge ...

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 must ...

but Dienes prevailed upon him to get involved in the theory of

integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

, thereby setting him on course for his life's work. A devoted

Methodist Methodism, also called the Methodist movement, is a group of historically related denominations of Protestant Christianity whose origins, doctrine and practice derive from the life and teachings of John Wesley. George Whitefield and John's b ...

, 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

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

PhD PHD or PhD may refer to: * Doctor of Philosophy (PhD), an academic qualification Entertainment * '' PhD: Phantasy Degree'', a Korean comic series * ''Piled Higher and Deeper'', a web comic * Ph.D. (band), a 1980s British group ** Ph.D. (Ph.D. albu ...

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 approximating the area of functions or lin ...

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 Göt ...

but which included the

Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Lebe ...

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 ( grc-gre, συμπόσιον ''symposion'' or ''symposio'', from συμπίνειν ''sympinein'', "to drink together") was a part of a banquet that took place after the meal, when drinking for pleasure was acc ...

in Real Analysis. His academic career began as Assistant Lecturer,

Bedford College for Women Bedford College was in York Place after 1874 Bedford College was founded in London in 1849 as the first higher education college for women in the United Kingdom. In 1900, it became a constituent of the University of London The University o ...

, 1947–48; then Assistant Lecturer at Birkbeck, 1948–51; Lecturer,

Queen's University Belfast , mottoeng = For so much, what shall we give back? , top_free_label = , top_free = , top_free_label1 = , top_free1 = , top_free_label2 = , top_free2 = , established = , closed = , type = Public research university , parent = ...

, 1951–56; Lecturer,

Bristol University , mottoeng = earningpromotes one's innate power (from Horace, ''Ode 4.4'') , established = 1595 – Merchant Venturers School1876 – University College, Bristol1909 – received royal charter , type ...

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

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

, 1964–70; Chair of Pure Mathematics,

New University of Ulster sco, Ulstèr Universitie , image = Ulster University coat of arms.png , caption = , motto_lang = , mottoeng = , latin_name = Universitas Ulidiae , established = 1865 – Magee College 1953 - Magee Un ...

, 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 serie ...

" (22, 1947); "The efficiency of matrices for

bounded sequences In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that :, f(x), \le M for all ''x'' in ''X''. A fun ...

" (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 societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

'', 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 cornerst ...

" ( 13, 1963); "Generalized integrals of vector-valued functions ( 19 1969) His additional papers were #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 that ...

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 ''Aequationes Mathematicae'' is a mathematical journal. It is primarily devoted to functional equations, but also publishes papers in dynamical systems, combinatorics, and geometry. As well as publishing regular journal submissions on these topics ...

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 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

Peng Yee Lee Peng Yee Lee (born 1938) is a Singaporean mathematician and mathematics educator. Lee is an associate professor of mathematics at the National Institute of Education in Singapore. He is a former president of the Southeast Asian Mathematical Socie ...

. 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.

References

* * * Whole Number 247 * {{DEFAULTSORT:Henstock, Ralph 1923 births 2007 deaths 20th-century English mathematicians 21st-century English mathematicians Mathematical analysts Academics of Queen's University Belfast Academics of the University of Bristol Academics of Lancaster University Academics of Ulster University Alumni of Birkbeck, University of London Alumni of St John's College, Cambridge English Methodists

Early life

Work

List of publications of Ralph Henstock

Review of Henstock's work

See also

References