mathematics

Paul Butzer

Paul Butzer is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Paul Butzer rather than just read about it. In short: Paul Leo Butzer (born 15 April 1928) is a German mathematician who specializes in Analysis (Approximation theory, Harmonic analysis). Life and work Butzer was born in Mülheim an der Ruhr on 15 April 1928.

Key takeaways

Paul Butzer belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Paul Butzer to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Paul Butzer from memory before moving on to harder problems.

Reference excerpt

Paul Leo Butzer (born 15 April 1928) is a German mathematician who specializes in Analysis (Approximation theory, Harmonic analysis).

Life and work Butzer was born in Mülheim an der Ruhr on 15 April 1928. He is the son of an engineer, and his mother studied mathematics at RWTH Aachen University. As opponents of the National Socialists (Nazis), Butzer's parents left Germany with their children in 1937 and moved to England. During World War II, they relocated to Canada, where Butzer attended school in Montreal and studied mathematics at Loyola College (later Concordia University), completing his bachelor's degree in 1948. He then pursued further studies at the University of Toronto, including studying under Harold Scott MacDonald Coxeter and William Tutte, and obtained his Ph.D. in 1951 under the supervision of George G. Lorentz (On Bernstein polynomials). In 1952, he became a lecturer and then an assistant professor at McGill University. In 1955/56, he lived in Paris and then in Mainz. He decided to stay in Germany, where he completed his habilitation at the University of Freiburg, taught in Würzburg, and starting in 1958, at RWTH Aachen University. In 1962, he was appointed professor there. In 1963, he began organizing international conferences on Approximation theory at the Oberwolfach Research Institute for Mathematics, later together with Béla Szőkefalvi-Nagy. In addition to approximation theory and its connections to Fourier analysis and semigroups of operators in Banach spaces, Butzer also worked on probability theory (Central limit theorem and related Convergence tests issues), Sampling Theory, and Signal analysis. Paul Leo Butzer also delved into the history of mathematics, particularly in its connection with Aachen. He studied figures such as Peter Gustav Lejeune Dirichlet, Eduard Helly, Eugène Catalan, Pafnuty Chebyshev, Charles Jean de la Vallée Poussin, the history of splines, Otto Blumenthal, mathematics in the Carolingian era, and Elwin Bruno Christoffel (on whom he published a book). Butzer is a member of the Royal Society of Sciences in Liège and the Royal Belgian Academy of Sciences. He is an honorary member of the Mathematical Society in Hamburg. He has received honorary doctorates from three universities: Liège, York, and Timișoara. Paul Butzer is the brother of Karl W. Butzer.

Publications With Hubert Berens: "Semi-groups of Operators and Approximation," Grundlehren der mathematischen Wissenschaften, Springer Verlag, 1967. With Hermann Schulte: "Ein Operatorenkalkül zur Lösung gewöhnlicher und partieller Differenzengleichungssysteme von Funktionen diskreter Veränderlicher und seine Anwendungen," Köln, Opladen, Westdeutscher Verlag, 1965. With Rolf Joachim Nessel: "Fourier Analysis and Approximation," Academic Press, Vol. 1 (One-dimensional Theory), 1971. With Walter Trebels: "Hilberttransformation, gebrochene Integration und Differentiation," Köln, Opladen, Westdeutscher Verlag, 1968. With Karl Scherer: "Approximationsprozesse und Intepolationsmethoden," BI Hochschultaschenbuch, Mannheim, 1968. With W. Oberdörster: "Darstellungssätze für beschränkte lineare Funktionale im Zusammenhang mit Hausdorff-, Stieltjes- und Hamburger-Momentenproblemen," Opladen, Westdeutscher Verlag, 1975. Edited with Dietrich Lohrmann: "Science in Western and Eastern Civilization in Carolingian Times," Birkhäuser, 1993. Edited with Walter Oberschelp and Max Kerner: "Karl der Grosse und sein Nachwirken: 1200 Jahre Kultur und Wissenschaft in Europa," 2 volumes, Turnhout, Brepols, 1997/98. "Mathematics in West and East from the fifth to tenth centuries: an overview," in P. L. Butzer, Dietrich Lohrmann (Eds.): "Science in Western and Eastern civilization in Carolingian times," Basel, 1993, pp. 443–481. With Karl W. Butzer: "Mathematics at Charlemagne's court and its transmission," in Catherine Cubitt (Ed.): "Court culture in the early middle ages," Turnhout, 2003, pp. 77–89. "The Mathematicians of the Aachen-Liège Region from the Carolingian to the Late Ottonian Period" (Die Mathematiker des Aachen-Lütticher Raumes von der karolingischen bis zur spätottonischen Epoche), in Annalen des Historischen Vereins für den Niederrhein, Volume 178, 1976, pp. 7–30. Edited with F. Féher: "E. B. Christoffel, the influence of his work on mathematics and the physical sciences," Birkhäuser, 1981. With Francois Jongmans: "P. L. Chebyshev (1821–1894): a guide to his life and work," Lehrstuhl für Mathematik A, RWTH Aachen, 1998. "Dirichlet and his role in the founding of mathematical physics," Lehrstuhl A für Mathematik, RWTH Aachen, 1983.

References

External links Homepage of the RWTH Aachen (German) O'Connor, John J.; Robertson, Edmund F., "Paul Butzer", MacTutor History of Mathematics Archive, University of St Andrews Mathematics Genealogy Project Author Profile in the zbMATH Databank

Illustrations

Paul Butzer: Butzer in Nice in 1970 — Butzer in Nice in 1970

Worked examples

Example 1 — a first encounter with Paul Butzer

Start with the simplest possible case. Write down what Paul Butzer claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Paul Butzer before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Paul Butzer ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Paul Butzer

In research: Paul Butzer appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Paul Butzer in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Paul Butzer is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1928 births, 20th-century German mathematicians, Academic staff of RWTH Aachen University, so understanding it makes those chapters shorter.
In everyday life: Look for Paul Butzer outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Paul Butzer in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paul Butzer means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Paul Butzer out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paul Butzer in simple terms?

Paul Leo Butzer (born 15 April 1928) is a German mathematician who specializes in Analysis (Approximation theory, Harmonic analysis). Life and work Butzer was born in Mülheim an der Ruhr on 15 April 1928.

Why does Paul Butzer matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Paul Butzer?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Paul Butzer.