Abstract
The article on Riemann derivatives by P. L. Butzer and W. Kozakiewicz of 1954 was the basis to generalizations of the classical scalar-valued derivatives to Taylor, Peano, and Riemann derivatives in the setting of semigroup theory. The present paper gives an overview of the 1954 article, describes its influence, and integrates it into the literature on related problems. It also describes the state of the mathematics department at McGill University where the article was written.
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Notes
For an orbituary of G. G. Lorentz see [33].
Erik Christopher Zeeman was born in Japan on 4 February 1925. His family moved to England 1 year after his birth. He studied mathematics at Christ’s College, Cambridge, and received an MA in 1950 and a PhD in 1954 (the latter under the supervision of Shaun Wylie) from the University of Cambridge. Zeeman was elected as a Fellow of the Royal Society in 1975 and was awarded the Society’s Faraday Medal in 1988. He was the 63rd President of the London Mathematical Society in 1986–88. He was awarded the Senior Whitehead Prize of the Society in 1982. Between 1988 and 1994 he was the Professor of Geometry at Gresham College. He received a knighthood in the 1991 Birthday Honors for “mathematical excellence and service to British mathematics and mathematics education” (extracted from [47], see also https://warwick.ac.uk/newsandevents/knowledgecentre/science/maths-statistics/zeeman and https://mathshistory.st-andrews.ac.uk/Biographies/Zeeman/).
References
Butzer, P.L., Kozakiewicz, W.: On the Riemann derivatives for integrable functions. Canadian J. Math. 6, 572–581 (1954). https://doi.org/10.4153/cjm-1954-062-5
Zygmund, A.: Trigonometrical series. Panstwowe Wydawnictwo Naukowe. Warszawa (1935). 2nd Edition: Dover Publications, New York (1955)
Brouwer, L.E.J.: Over differentiequotienten en differentiaalquotienten. Amst. Ak. Versl. 17, 38–45 (1908)
Popoviciu, T.: Sur les solutions bornees et les solutions mesurables de certaines équations fonctionnelles. Mathematica, Cluj 14, 47–106 (1938)
Butzer, P.L., Nessel, R.J.: Fourier analysis and approximation. Birkhäuser, Basel; Academic Press, New York (1971)
Verblunsky, S.: The generalized third derivative and its application to the theory of trigonometric series. Proc. London Math. Soc. s1-31(1), 387–406 (1930). https://doi.org/10.1112/plms/s2-31.1.387
Verblunsky, S.: The generalized fourth derivative. J. London Math. Soc. s1-6(2), 82–84 (1931). https://doi.org/10.1112/jlms/s1-6.2.82
Saks, S.: On the generalized derivatives. J. London Math. Soc. s1-7(4), 247–251 (1932). https://doi.org/10.1112/jlms/s1-7.4.247
Dutta, T.K., Mukhopadhyay, S.N.: On the Riemann derivatives of \({\rm C}_s{\rm P}\)-integrable functions. Anal. Math. 15(3), 159–174 (1989). https://doi.org/10.1007/BF02020765
Mitra, S., Mukhopadhyay, S.N.: Convexity conditions for generalized Riemann derivable functions. Acta Math. Hungar. 83(4), 267–291 (1999). https://doi.org/10.1023/A:1006696218988
Mukhopadhyay, S.N.: Higher order derivatives. Chapman & Hall/CRC, Boca Raton, FL (2012). In collaboration with P. S. Bullen
Kassimatis, C.: Functions which have generalized Riemann derivatives. Canad. J. Math. 10, 413–420 (1958). https://doi.org/10.4153/CJM-1958-040-x
Burkill, J.C.: The Cesáro-Perron scale of integration. Proc. Lond. Math. Soc. 2(39), 541–552 (1935). https://doi.org/10.1112/plms/s2-39.1.541
Kemperman, J.H.B.: On the regularity of generalized convex functions. Trans. Amer. Math. Soc. 135, 69–93 (1969). https://doi.org/10.1090/S0002-9947-1969-0265531-3
Butzer, P.L., Berens, H.: Semi-groups of operators and approximation. Springer, New York (1967)
Butzer, P.L., Tillmann, H.G.: Approximation theorems for semi-groups of bounded linear transformations. Math. Ann. 140, 256–262 (1960)
Butzer, P.L.: Beziehungen zwischen den Riemannschen, Taylorschen und gewöhnlichen Ableitungen reellwertiger Funktionen. Math. Ann. 144, 275–298 (1961)
Berens, H., Westphal, U.: Zur Charakterisierung von Ableitungen nichtganzer Ordnung im Rahmen der Laplace-Transformation. Math. Nachr. 38, 115–129 (1968)
Berens, H., Westphal, U.: A Cauchy problem for a generalized wave equation. Acta Sci. Math. (Szeged) 29, 93–106 (1968)
Balakrishnan, A.V.: Fractional powers of closed operators and the semigroups generated by them. Pacific J. Math. 10, 419–437 (1960)
Westphal, U.: Fractional powers of infinitesimal generators of semigroups. In: Hilfer, R. (ed.) Applications of Factional Calculus in Physics, pp. 131–170. World Sci. Publ., River Edge, NJ (2000). https://doi.org/10.1142/9789812817747_0003
Butzer, P.L., Westphal, U.: An access to fractional differentiation via fractional difference quotients. In: Ross, B. (ed.) Fractional Calculus and Its Applications (Proc. Internat. Conf., Univ. New Haven, West Haven, Conn., 1974). Lecture Notes in Math., vol. 457, pp. 116–145. Springer, Berlin (1975)
Butzer, P.L., Westphal, U.: An introduction to fractional calculus. In: Hilfer, R. (ed.) Applications of Fractional Calculus in Physics, pp. 1–85. World Sci. Publ., River Edge, NJ (2000). https://doi.org/10.1142/9789812817747_0001
Butzer, P.L., Stens, R.L.: A retrospective on research visits of Paul Butzer’s Aachen research group to Eastern Europe and Tenerife. Sampl. Theory Signal Process. Data Anal. 20(2), (2022). https://doi.org/10.1007/s43670-022-00034-6. Id/No 17
Lions, J.-L., Magenes, E.: Non-homogeneous boundary value problems and applications. vol. I, II, III, Springer, New York-Heidelberg (1972/73)
Görlich, E., Nessel, R.J.: Über Peano- und Riemann-Ableitungen in der Norm. Arch. Math. (Basel) 18, 399–410 (1967)
Fejzić, H.: On generalized Peano and Peano derivatives. Fund. Math. 143(1), 55–74 (1993)
Banach, S.: Theory of linear operations. Elsevier (North-Holland), Amsterdam (1987). Translated from the French by F. Jellett
Kühn, F., Schilling, R.L.: For which functions are \(f(X_t)-\mathbb{E} f(X_t)\) and \(g(X_t)/\mathbb{E} g(X_t)\) martingales? Theory Probab. Math. Statist. 105, 79–91 (2021). https://doi.org/10.1090/tpms
Krysko, M.: Wacław Kozakiewicz (1911–1959). Przegląd Statyst. 61(2), 207–209 (2014)
W.L.G.W.: Waclaw Kozakiewicz. In memoriam. Canad. Math. Bull. 2(2), 148–150 (1959). https://doi.org/10.1017/S0008439500025236
Plesken, W.: Hans Zassenhaus: 1912–1991. Jahresber. Deutsch. Math.-Verein. 96(1), 1–20 (1994)
De Boor, C., Nevai, P.: In memoriam: George G. Lorentz (1910–2006). J. Approx. Theory 156(1), 1–27 (2009). https://doi.org/10.1016/j.jat.2006.10.009
Butzer, P.L., Wickeren, E.: Book review: Moduli of smoothness by Z. Ditzian and V. Totik. Bull. Amer. Math. Soc. (N.S.) 19(2), 568–572 (1988)
Butzer, P.L.: Dominated convergence of Kantorovitch polynomials in the space \(L^p\). Trans. Roy. Soc. Canada Sect. III(46), 23–27 (1952)
Butzer, P.L.: Linear combinations of Bernstein polynomials. Canadian J. Math. 5, 559–567 (1953). https://doi.org/10.4153/cjm-1953-063-7
Butzer, P.L.: On two-dimensional Bernstein polynomials. Canad. J. Math. 5, 107–113 (1953). https://doi.org/10.4153/cjm-1953-014-2
Butzer, P.L.: On the extensions of Bernstein polynomials to the infinite interval. Proc. Amer. Math. Soc. 5, 547–553 (1954). https://doi.org/10.2307/2032032
Higgins, J.R., Stens, R.L. (eds.): Sampling theory in Fourier and signal analysis. vol. 2: Advanced Topics. Oxford University Press, Oxford (1999)
Butzer, P.L., Stens, R.L.: A retrospective on research visits of Paul Butzer’s Aachen research group to the Middle East, Egypt, India and China (in preparation) (2024)
Dodson, M.M.: Approximating signals in the abstract. Appl. Anal. 90(3–4), 563–578 (2011). https://doi.org/10.1080/00036811003627575
Butzer, P.L., Higgins, J.R., Stens, R.L.: Sampling theory of signal analysis. In: Pier, J.-P. (ed.) Development of mathematics 1950–2000, pp. 193–234. Birkhäuser, Basel (2000)
Butzer, P.L., Dodson, M.M., Ferreira, P.J.S.G., Higgins, J.R., Schmeisser, G., Stens, R.L.: Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections. Bull. Math. Sci. 4(3), 481–525 (2014). https://doi.org/10.1007/s13373-014-0057-3
Butzer, P.L., Stens, R.L.: A retrospective on research visits of Paul Butzer’s Aachen research group to North America and Western Europe. J. Approx. Theory 257 (2020). https://doi.org/10.1016/j.jat.2020.105452. Id/No 105452
Butzer, P.L., Lohrmann, D. (eds.): Science in western and eastern civilization in Carolingian times. Birkhäuser, Basel (1993)
Federwisch, M., Dieken, M.L., Meyts, D. (eds.): Insulin & related proteins – structure to function and pharmacology (contributions presented at the Alcuin Symposium, held at RWTH Aachen, April 2000). Kluwer Academic Publishers, New York (2002)
Rand, D.A.: Sir Erik Christopher Zeeman. 4 February 1925–13 February 2016. Biogr. Mems Fell. R. Soc. 73, 521–547 (2022). https://doi.org/10.1098/rsbm.2022.0012
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The authors are grateful to the two referees for their suggestions on how to reorganize the content of the paper, resulting in a clearer presentation.
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Communicated by: Tomas Sauer
A tribute to Maurice Dodson, a unique and long-standing friend of the authors (A short biography of Maurice Dodson can be found as Appendix C)
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Appendices
Appendix A: Paul Butzer’s reminiscences of the mathematics department at McGill University from 1945 to 1959 and the spirit of the collaboration with W. Kozakiewicz
In 1951, when Professor Wacław Kozakiewicz came to McGill University, Montréal, he was highly respected in Canada as an expert, versatile mathematician, not only in his research area, mathematical statistics but also in probability and real variable theory. His foregoing stations had been the Dominion Bureau of Statistics (\(={}\)DBS), Ottawa), the University of Sasketchevan (Saskatoon), and the Université de Montréal; see [30, 31] and Appendix B.
When McGills’ Principal Dr. Frank James and Dr. Herbert Tate McGills Chairman of Mathematics (from ca. 1935 to 1964?) decided to create a graduate school in 1945, their chief aim was to attract first-class mathematicians of senior rank. The first to come was Hans Zassenhaus from the University of Hamburg, who was appointed Peter Redpath Professor of Mathematics in 1949 (my brother Karl Butzer received his BSc in mathematics in 1954, Dr. Zassenhaus being the advisor); cf. [32]. The mathematical analyst Charles Fox, from Birkbeck College, London, since 1949 at McGill, who was promoted to professor in 1956, was a pleasant colleague. He raised the question whether there exist processes which simultaneously approximate and interpolate a given function. Such a process is the Fejèr-Hermite interpolation process, not aware to approximation theorists in North America at the time.
While Wacław came in 1951, Philip R. Wallace, a doctoral student of Leopold Infeld at Toronto, was a theoretical physicist and one of the bright lights of the mathematics department already since 1946.
After Dr. Tate had heard that Dr. Zassenhaus had offered me a position in his research group in spring 1952 (I had received my PhD in mathematics, minor physics, at the University of Toronto in 1951, Dr. George Lorentz being the advisor), he counter-offered with a regular lectureship (with promotion to Assistant Professor in 1953). During my 3 years at McGill I gave graduate courses on the theory of divergent series (Dr. Loyd Williams, who had taught it regularly, passed this course on to me before his retirement 1954) and Lebesgue integration (Jean Maranda being one of my good students; he received his PhD under Dr. Zassenhaus).
Until 1945, mathematics had been almost wholly a service department, mainly for engineering, with only seven faculty members. Dr. Zassenhaus had alone nine PhD students during his 10 years at McGill, seven until 1954. The first in 1950 was Joachim Lambek (1922–2014), who was born in Leipzig and spent 2 years in a Canadian internment camp. He was the first PhD in mathematics granted at McGill, even the first in the Province of Quebec. (The total number of students and PhDs in mathematics was not available to me for the period.)
All in all, the establishment of a graduate school in mathematics was a full success. For the year 1964, it is known to have some 40 newcomer students in the graduate school per year, half of them masters, the other half being doctors. According to the “QS The World University Ranking,” released June 2023, McGill ranks 30 (Toronto 21).
This was the state of mathematics when Wacław and I wrote our joint paper in late fall of 1952. At the time Wacław was a well-established mathematician in Canada, a product of the unique generation of mathematicians born in Poland between the two World Wars, with nine papers to his credit, in the broad area of probability theory (the first of 1933), while I had just two on approximation theory; he was 41, I just 24, a youngster in the Canadian mathematical community. We worked in Wacław’s home in Montréal-Westmount, while his spouse kindly prepared tea or hot chocolate with sandwiches or cakes. She was a refined, reserved, but friendly lady, who was always concerned that Wacław may excerpt himself too much; he seemed to have health problems. They had a son, John (Christopher), who often wanted to play with us while we were working.
Wacław, who was a polite, contemplative, very helpful, and kind person, never made me feel like a youngster during our many sessions. We tackled a problem, new for us two, in close cooperation.
In contrast, Dr. LorentzFootnote 1 (who claimed that through his mother, who was a member of the large Prince Chergodaev family, he was a descendant of Genghis Khan the fearsome Mongol warrior of the 13th century) had suggested a thesis topic, namely generalizing a theorem of A. O. Gelfond in a complex function theory setting. I saw no way and have not seen such a generalization since. There was no constructive help on his part—just criticism. I thought of my own topics; the approach of one of them, on linear combinations of Bernstein polynomials, was applied by some dozens of authors to a variety of approximation processes, in particular under the heading “linear combinations”; see [34]. The papers [35,36,37] were chapters of my Toronto dissertation. [38] was completed shortly after them. It is quite often cited in the more recent literature.
Let me emphasize that the kind, helpful, and cooperative way Wacław worked with me became the model for my work with my many students I later had in Aachen.
Appendix B: A short biography of Wacław Kozakiewicz
Wacław Kozakiewicz was born in Warsaw on 23 January 1911, and baptized in the village Złotniki in the Jędrzejów district. He was the son of Jan Kozakiewicz (1882–1945) and Taida Maria Anna, née Trószyńska. His brother was Stefan Kozakiewicz, later professor of art history at Warsaw University.
From September 1920 he was a pupil at the Mikołaj-Rey Grammar School in Warsaw, where his father was mathematics and physics teacher and director; see https://www.rej.edu.pl/dyrektorzy_szkoly/. He received his school-leaving certificate (humanistic type) in May 1929.
In October 1929, he began his studies of mathematics at the University of Warsaw. At first he was interested in the theory of analytical functions, then in probability theory and mathematical statistics. He attended lectures by Wacław Sierpiński, Jan Łukasiewicz, Stanisław Saks, Otto Nikodým and Aleksander Rajchman, among others, but was mainly a student of Stefan Mazurkiewicz. He worked in the student Mathematical and Physical Circle and was its vice-president for 1 year. On 7 November 1933, he was awarded the Master’s degree.
In the academic year 1933/34, he studied at the Faculty of Humanities of the University of Warsaw as part of the Pedagogical Year Study.
From November 1933 to October 1938 he was a senior assistant in the Department of Mathematical Statistics at the Faculty of Horticulture of the Warsaw University of Life Sciences (SGGW) in Warsaw, headed by the statistician Jerzy Spława-Neyman. He received his doctorate in mathematics from the University of Warsaw in 1936, his thesis being supervised by Stefan Mazurkiewicz.
At the onset of the war, he joined the Polish armed forces in France and served until the surrender of France in June 1940. As he could not leave France, he taught mathematics at the Polish lyceum for the next 4 years. When he was ordered to report for forced labor in Germany, he fled via Spain to Canada, where he arrived in October 1944.
There he first joined the staff of the Dominion Bureau of Statistics (DBS) in Ottawa, and in the autumn of 1945, he became an associate professor of mathematics at the University of the Province of Saskatchewan in Saskatoon, Canada, where he became a close friend of W. J. R. Crosby, who valued him most highly. In 1949, the Université de Montréal wanted to establish a program in mathematical statistics and they were therefore looking for a mathematician trained in the French tradition. They offered Kozakiewicz this position, which he accepted. In 1951 he migrated to McGill University.
He died of a heart attack in Montréal on 8 March 1959, leaving his wife Naomi, née Pelletier (who died in her home on March 27, 2014, at the age of 102), a Canadian, and their son John. He had met Naomi at the DSB, they were married in the winter of 1947.
This biography is based on that of Krysko [30] and the obituary [31], which also contains a list of his publications.
Appendix C: A short biography of Maurice Dodson
Michael Maurice Dodson and his twin brother George Guy Dodson were born in Palmerston North, New Zealand on 1 January 1937. Maurice graduated from Auckland University in 1957 with a BSc, and in 1958 with an MSc in Maths (1st Class Honors) and the Mathematics Prize. He went to Cambridge in 1959 and graduated with a BA in Maths in 1962. He studied Number Theory under Harold Davenport and obtained his PhD in additive Number Theory in 1965. His brother Guy studied chemistry and went on to Oxford in 1961 to work with the eminent Professor Dorothy Hodgkin, who was awarded a Nobel Prize in chemistry in 1964.
Maurice joined the just 1-year-old University of York in 1964. His research interests broadened from Diophantine equations and approximation in number theory to catastrophe theory, and through a connection with X-ray crystallography, came to include Fourier series, harmonic analysis, and applications to sampling in 1985 when he first met Rowland Higgins. He also had become interested in chaos, biology, and dynamical systems. According to MathSciNet, Maurice was the author of 90 scientific publications in these fields.
As for sampling, Maurice focused on sampling in abstract spaces. In this regard, see his chapter “Abstract harmonic analysis and the sampling theorem” in [39, Chapter 10], written together with M. G. Beaty. He was an invited speaker at the SampTA conference held at Samsun, Turkey, 2005, with the lecture “The Whittaker-Kotel’nikov-Shannon sampling theorem in abstract harmonic spaces”; see [40]. He contributed to the workshop “Approximation Theory and Signal Analysis” in Lindau 2009 to celebrate Paul Butzer’s 80th year with “Approximating signals in the abstract” [41]. See also [42, Section 11.2] and [43, Sections 6.2 and 6.3].
Maurice was invited to the British-Russian Workshop in Functional Analysis, held at the Euler International Mathematical Institute (associated with the Steklov Institute), St. Petersburg, from 13 to 17 October 1996. There were about 30 participants from different parts of the former Soviet Union and from England, Scotland, and Wales.
In 1988, the authors participated in Maurice’s “Symposium on Fourier Analysis, Interpolation and Signal Processing” in York. A large number of prominent mathematicians were present, including Walter Hayman from Imperial College, who had invited me (PLB) to a lecture at his university during my first trip to England and Scotland in 1973, together with my parents, a tour organized by Lionel Cooper. Jointly with Jim Clunie, Maurice organized an even larger follow-up conference in 1993.
During the 1988 symposium, Maurice suggested that the University of York and RWTH Aachen join forces in establishing the Aachen-York “Alcuin Symposia,’’ a series of conferences on mathematics, history, electrical engineering, and on biochemistry. It was Johannes Erger, chairman of the external institute of the RWTH (see [24]) who succeeded in incorporating its core into the Erasmus Student Mobility program of the EU, a student exchange program that ultimately involved 22 universities from seven EU countries. Many students from Aachen and most of those going from the Lehrstuhl A für Mathematik, went to the University of York, especially because Maurice took care of them; see also [44]. In connection with the Erasmus program, he took part in workshops in Segovia (Spain), Orleans (France), Thessaloniki (Greece), and Assisi (Italy).
The first of the “Alcuin symposia,’’ conducted by the authors, was held in Aachen in 1989, followed by one in York 1 year later and the symposium “Science and History in Western and Eastern Civilisation’’ in Aachen in 1991; the proceedings appeared in [45]. The Alcuin Symposia on biochemistry were held as joint workshops on insulin and related proteins between the groups of Guy Dodson in York and the groups of Dietrich Brandenburg and Axel Wollmer in Aachen. The last one took place in Aachen in 2000; for the proceedings see [46].
Maurice was due to retire in 2004, but the closure of the Department of Mathematics at the nearby University of Hull, announced in late in 2004, led the two Universities agreeing that the York Department would adapt and take on those whom it could. This was a very complicated and delicate operation at a time when York, like many universities in the UK, was also experiencing cuts. Moreover, the Head of Department was due for a year’s leave. In view of his competence and international reputation, the Vice-Chancellor appealed to Maurice to postpone his retirement and serve as Head of Department for a year. The many problems of staff and student accommodation were managed by Maurice, the new program started on time and Maurice finally retired for good, now as Professor Emeritus in 2005.
In 1974, he married Haleh Afshar, an Iranian who became a professor of politics and women’s studies at the University of York, was awarded an OBE for her work on women and Islam and was elevated to the House of Lords in 2008, as Baroness Haleh Afshar of Heslington. She died too early on 12 May 2022.
At a convocation ceremony at the University of York in July 1997, three persons were honored with a Doctorate from the University, the retired Archbishop of York, John Habgood, the renowned president of the European Commission, Jacques Delors, and I (PLB) myself; see [44]. Maurice proposed and organized in full the honorary doctorates for Christopher ZeemanFootnote 2 and myself, and together with his Number Theory colleagues Richard Hall and Terence Jackson that for Paul Erdős. My deep thanks are due to Maurice for this special honor.
Rudolf and Paul always have fond memories of their visits to York, where they were often invited by Maurice to his cozy house and hosted by him and his wife Haleh at a huge wooden table with extraordinary warmth; see also [44]. Rudolf and Paul would like to express their profound gratitude to Maurice for his true friendship over the many years.
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Butzer, P.L., Stens, R.L. The Butzer-Kozakiewicz article on Riemann derivatives of 1954 and its influence. Adv Comput Math 49, 86 (2023). https://doi.org/10.1007/s10444-023-10086-4
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DOI: https://doi.org/10.1007/s10444-023-10086-4