{{Short description|Hungarian-American mathematician (1893–1974)}} {{Redirect|Lanczos|resampling method|Lanczos resampling}} {{Infobox scientist | name = Cornelius Lanczos | image = File:Lanczos Kornel photo in 1947.jpg | image_size = | alt = | caption = Lanczos in 1947 | birth_date = {{birth date|1893|02|02}} | birth_place = Székesfehérvár, Austria-Hungary | death_date = {{death date and age|1974|06|25|1893|02|02}} | death_place = Budapest, Hungary | fields = Mathematics<br/>Theoretical physics | workplaces = {{ubil|University of Freiburg | Frankfurt University | Purdue University | Boeing | National Bureau of Standards | Dublin Institute for Advanced Studies}} | alma_mater = University of Budapest<br/>University of Szeged | thesis_title = Relation of Maxwell's Aether Equations to Functional Theory | thesis_year = 1921 | doctoral_advisor = Rudolf Ortvay | academic_advisors = Loránd Eötvös<br/>Lipót Fejér<br/>Erwin Madelung | doctoral_students = | notable_students = | known_for = {{ubil|Lanczos algorithm| Lanczos tensor| Lanczos resampling| Lanczos approximation| Lanczos sigma factor| Lanczos differentiator| Lanczos–van Stockum dust}} | influences = | influenced = | awards = Chauvenet Prize (1960)<ref>{{cite journal|author=Lanczos, Cornelius|title=Linear Systems in Self-Adjoint Form|journal=Amer. Math. Monthly|volume=65|year=1958|issue=9 |pages=665–679|url=http://www.maa.org/programs/maa-awards/writing-awards/linear-systems-in-self-adjoint-form|doi=10.2307/2308707|jstor=2308707 |url-access=subscription}}</ref> | spouse = {{ubil|Mária Erzsébet Rump (1928–1939)|Ilse Hildebrand (1954–1974)}} | footnotes = }} '''Cornelius (Cornel) Lanczos''' ({{langx|hu|Lánczos Kornél}}, {{IPA|hu|ˈlaːnt͡soʃ ˈkorneːl|pron}}; born as '''Kornél Lőwy''', until 1906: ''Löwy (Lőwy) Kornél''; February 2, 1893 – June 25, 1974) was a Hungarian, American, and later Irish mathematician and physicist. According to György Marx he was one of the Martians,<ref name=":3">{{Cite book |last=Marx |first=George |title=The Voice of the Martians |publisher=Akadémiai Kiadó |year=2001 |isbn=978-9-630-57830-1 |edition=3rd revised |location=Budapest |chapter=Cornelius Lanczos}}</ref> a group of Hungarian scientific luminaries who immigrated to the United States to escape national socialism. He was remembered by his colleagues as an innovative scholar and an excellent educator.<ref name=":4" /><ref name=":5" />
==Early life and education== He was born in Fehérvár (Alba Regia), Fejér County, Kingdom of Hungary, Austria-Hungary<ref name=":0">{{Cite journal |last=Yourgrau |first=Wolfgang |date=1975 |title=Cornelius Lanczos (1893-1974) |url=https://link.springer.com/article/10.1007/BF01100311 |journal=Foundations of Physics |volume=5 |issue=1 |pages=19–20 |doi=10.1007/BF01100311|bibcode=1975FoPh....5...19Y |url-access=subscription }}</ref> to Károly Lőwy and Adél Hahn. He grew up in relative comfort and attended a Catholic Gymnasium (high school). Between 1911 and 1916, he studied at the University of Budapest, where one of his professors in physics was Roland Eötvös, whose skills as an experimental physicist impressed him.<ref name=":4">{{Cite journal |last1=Yodin |first1=Erwin Y. |last2=Butler |first2=R. |last3=Erdélyi |first3=A. |last4=Gellai |first4=B. |last5=McConnell |first5=J. R. |last6=Ortiz |first6=Eduardo L. |last7=Rhodes |first7=I. |date=1975 |title=In memory of Cornelius Lánczos |url=https://www.sciencedirect.com/science/article/pii/0898122175900243/pdfft?md5=9f703de9e1ef32b2deeaff15828b900d&pid=1-s2.0-0898122175900243-main.pdf |journal=Computers and Mathematics with Applications |volume=1 |issue=3 |pages=257–268 |doi=10.1016/0898-1221(75)90024-3}}</ref> In mathematics, his notable teacher was Lipót Fejér, then a young mathematician.<ref name=":4" /> Lanczos graduated with a teacher's diploma in mathematics and physics. He worked an assistant of Károly Tangl at the Department of Experimental Physics at the Polytechnical University of Budapest from 1916 to 1921.<ref name=":4" />
In his doctoral dissertation titled ''The Relation of Maxwell's Aether Equations to Functional Theory'',<ref>{{cite arXiv |eprint=physics/0408079 |class= |first=Cornelius |last=Lanczos |title=The relations of the homogeneous Maxwell's equations to the theory of functions |authorlink= |year=2004}}</ref> Lanczos re-wrote Maxwell's equations of electromagnetism in terms of quaternions and applied a relativistic variational principle.<ref name=":3" /> He sent a copy of his thesis to Albert Einstein, who replied, "I studied your paper as far as my present overload allowed. I believe I may say this much: this does involve competent and original brainwork, on the basis of which a doctorate should be obtainable... I gladly accept the honorable dedication."<ref name=":2">{{Cite book |last=Gellai |first=Barbara |title=The Intrinsic Nature of Things: The Life and Science of Cornelius Lanczos |publisher=American Mathematical Society |year=2010 |isbn=978-0-821-85166-1}}</ref>{{rp|20}} Lanczos maintained his contact with Einstein for another 35 years, until the latter's death.<ref name=":3" /> In 1921, Lanczos completed his Ph.D. training at the University of Szeged under the supervision of Rudolf Ortvay, a former student of Arnold Sommerfeld.<ref name=":3" /> While Ortvay was not distinguished as a researcher, he was an inspirational teacher who brought modern physics to Hungary.<ref name=":3" /><ref name=":4" />
== Career == As a consequence of the restrictions from the new right-wing regime in Hungary, Lanczos moved to Germany in search of employment.<ref name=":3" /> From 1921 to 1924, Lanczos served as a lecturer at the University of Freiburg.<ref name=":0" /> In 1924 he discovered an exact solution to the Einstein field equations of general relativity representing a cylindrically symmetric rigidly rotating configuration of dust particles.<ref>{{Cite journal |last=Lanczos |first=Cornelius |date=1924 |title=Über eine stationäre Kosmologie im Sinne der Einsteinschen Gravitationstheorie |trans-title=On a static cosmology in the sense of Einstein's theory of gravity |url=https://link.springer.com/article/10.1007/BF01328251 |journal=Zeitschrift für Physik |language=German |volume=21 |issue=1 |pages=73–110 |bibcode=1924ZPhy...21...73L |doi=10.1007/BF01328251 |url-access=subscription}}</ref> This was later rediscovered by Willem Jacob van Stockum in 1938.<ref>{{Cite journal |last=van Stuckum |first=Willem Jacob |date=1938 |title=The gravitational field of a distribution of particles rotating around an axis of symmetry |journal=Proceedings of the Royal Society of Edinburgh |volume=57 |pages=135–154 |doi=10.1017/S0370164600013699}}</ref> It is one of the simplest known exact solutions in general relativity<ref>{{Cite journal |date=February 2, 2015 |title=Cornelius Lanczos |url=https://pubs.aip.org/physicstoday/online/8307 |journal=Physics Today |issue=2 |article-number=8307 |doi=10.1063/PT.5.030887|bibcode=2015PhT..2015b8307. |url-access=subscription }}</ref> and is regarded as an important example, in part because it exhibits closed timelike curves.<ref>{{Cite web |last=Beenakker |first=Carlo |author-link=Carlo Beenakker |date=2004 |title=Time travel pioneer |url=https://ilorentz.org/history/stockum/stockum.html |website=Lorentz Institute}}</ref>
Lanczos worked at the University of Frankfurt from 1924 to 1931,<ref name=":0" /> delivering lectures for Erwin Madelung as a Privatdozent.<ref name=":4" />{{Reference page|page=491}} He also briefly served as assistant to Albert Einstein in Berlin during the academic year 1928–29,<ref name=":2" />{{rp|27}}upon invitation by the latter.<ref name=":0" /> It was Leo Szilard who recommended him to Einstein.<ref name=":3" /> Einstein wrote to Madelung, requesting a leave of absence for Lanczos, which was granted. Before leaving for Berlin, Lanczos wrote to Einstein that Hans Bethe was being considered as his temporary replacement.<ref name=":02">{{cite book |last=Pais |first=Abraham |author-link=Abraham Pais |title='Subtle is the Lord..': The Science and the Life of Albert Einstein |title-link=Subtle is the Lord |publisher=Oxford University Press |year=1982 |isbn=0-19-853907-X |location=Oxford |pages=}}</ref>{{Reference page|page=491}} By the time he went to work with Einstein, Lanczos had already written multiple papers on relativity.<ref name=":02" />{{Reference page|page=491}} In Berlin, Lanczos examined the motion of singularities—meaning, particles—in curved spacetime as described by general relativity. Einstein had a high opinion of Lanczos for his mathematical skills. In this capacity, Lanczos replaced Marcel Grossmann as Einstein's collaborator, helping him with the difficult mathematics of general relativity.<ref name=":3" /> Although Einstein and Lanczos published no papers together, Einstein referred to the works of Lanczos in one of his subsequent articles on distant parallelism.<ref name=":02" />{{Reference page|page=491}} Another innovation of Lanczos in this context was harmonic coordinates, which he introduced independently of Théophile De Donder. These were later used by Yvonne Choquet-Bruhat in her proof of the local existence and uniqueness of solutions to the Einstein field equations in vacuum with initial values and in algorithms of numerical relativity that simulate the spiral and merger of binary systems of compact objects, which emit gravitational waves.<ref name=":7">{{Cite journal |last=Bieri|first=Lydia|date=March 2020|title=Book Review: ''A Lady Mathematician in this Strange Universe: Memoirs''|journal=Notices of the American Mathematical Society|volume=67|issue=3|pages=384–9|doi=10.1090/noti2055}}</ref>
Following the seminal publication of Werner Heisenberg announcing the creation of his matrix formulation of quantum mechanics in 1925,<ref>{{Cite journal |last=Heisenberg |first=Werner |date=1925 |title=Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen |trans-title=Quantum-Theoretical Re-interpretation of Kinematic and Mechanical Relations |url=https://en.wikipedia.org/wiki/%C3%9Cber_quantentheoretische_Umdeutung_kinematischer_und_mechanischer_Beziehungen |journal=Zeitschrift für Physik |language=German |volume=33 |pages=879–893 |doi=10.1007/BF01328377 |bibcode=1925ZPhy...33..879H }}</ref> Lanczos wrote a paper demonstrating how the new theory could be expressed in terms of linear integral equations.<ref>{{Cite journal |last=Lanczos |first=Cornelius |date=1926 |title=Über eine feldmaßifssige Darstellung der neuen Quantenmechanik |trans-title=On a Field-based Representation of the New Quantum Mechanics |url=https://link.springer.com/article/10.1007/BF01379857 |journal=Zeitschrift für Physik |language=German |volume=35 |pages=812–30 |doi=10.1007/BF01379857|url-access=subscription }}</ref><ref name=":3" /> However, at the time, this paper had little impact, in part because physicists were more used dealing with differential equations.<ref name=":12" />{{Reference page|page=276}} Erwin Schrödinger published a series of papers detailing his own undulatory version of quantum theory, which proved rather popular among physicists.<ref name=":3" /> But Lanczos' paper made it clear that the two seemingly different formulations of quantum mechanics were in fact equivalent,<ref name=":3" /> something Schrödinger himself later proved.<ref>{{Cite journal |last=Schrödinger |first=Erwin |date=1926 |title=Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen |trans-title=On the Relationship between the Heisenberg-Born-Jordan Quantum Mechanics and My Own |journal=Annalen der Physik |language=German |volume=79 |issue=8 |pages=734–56 |doi=10.1002/andp.19263840804 |bibcode=1926AnP...384..734S }}</ref> Carl Eckart independently reached the same conclusion, based on the work of Lanczos.<ref>{{Cite journal |last=Eckart |first=Carl |date=1926 |title=The Solution of the Problem of the Simple Oscillator by a Combination of the Schroedinger and the Lanczos Theories |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=12 |issue=7 |pages=473–6 |doi=10.1073/pnas.12.7.473|doi-access=free |pmid=16587109 |pmc=1084628 |bibcode=1926PNAS...12..473E }}</ref><ref>{{Cite journal |last=Eckart |first=Carl |date=1926 |title=Operator Calculus and the Solution of the Equations of Quantum Dynamics |url=https://journals.aps.org/pr/abstract/10.1103/PhysRev.28.711 |journal=Physical Review |volume=28 |issue=4 |pages=711–26 |doi=10.1103/PhysRev.28.711 |bibcode=1926PhRv...28..711E }}</ref> This paper also helped Paul Dirac create his own formulation of quantum mechanics as a theory of linear transformations.<ref name=":12">{{Cite book |last=Jammer |first=Max |author-link=Max Jammer |title=The Conceptual Development of Quantum Mechanics |publisher=McGraw-Hill |year=1966 |isbn=978-0-88318-617-6}}</ref>{{Reference page|pages=300-1}} Lanczos's 1926 paper was the earliest continuum-theoretic formulation of quantum mechanics;<ref name=":4" /> it was close to the notion of a quantum field.<ref name=":12" />{{Reference page|page=276}} Moreover, Lanczos was willing to accept the probabilistic interpretation of the wave function.<ref name=":5">{{Cite web |last=Schulte |first=Tom |date=December 31, 2010 |title=Book Review: ''The Intrinsic Nature of Things: The Life and Science of Cornelius Lanczos'' |url=https://old.maa.org/press/maa-reviews/the-intrinsic-nature-of-things-the-life-and-science-of-cornelius-lanczos |access-date=December 19, 2025 |website=MAA Reviews |publisher=Mathematical Association of America}}</ref> In 1972, at an event organized by the European Physical Society in Trieste, Italy, Bartel Leendert van der Waerden publicly recognized the significance of that paper, which correctly formulated the eigenvalue problem in terms of integration and even came close to introducing the Dirac <math>\delta</math>-distribution. But van der Waerden was unaware that Lanczos was in the audience until Léon Rosenfeld urged the latter to come to the stage.<ref name=":4" />
In 1927 Lanczos married Maria Rupp.<ref name=":2" />{{rp||pages=41, 53}} He moved to the United States in 1931.<ref name=":4" /> Mindful of the Great Depression, he turned his attention towards applied mathematics.<ref name=":3" /> He began conducting research in numerical analysis,<ref name=":4" /> and developed a number of concepts in service of early digital computers.<ref name=":5" /> He served as a professor of mathematics and aeronautical engineering at Purdue University from 1931 to 1946.<ref name=":0" /> Between 1927 and 1939, Lanczos split his life between two continents. His wife Maria Rupp, who had contracted tuberculosis, stayed with Lanczos' parents in Székesfehérvár year-around while Lanczos went to Purdue for half the year, teaching graduate students matrix mechanics and tensor analysis.<ref name=":2" />{{rp||pages=41, 53}} His lecture notes on quantum mechanics examined in detail its mathematical formulation, including topics in function space and group theory.<ref name=":3" /> At Purdue, he introduced an "experimental" curriculum for female students.<ref name=":5" />
In 1933 his son Elmar was born; Elmar came to Lafayette, Indiana with his father in August 1939, just before the Second World War broke out.<ref name=":2" />{{rp||pages=41, 53}} Maria died in 1938, the same year Lanczos became an American citizen. His father died the following year.<ref name=":3" /> After the War, he left Purdue<ref name=":4" /> and moved to Seattle, working for the Boeing Aircraft Company and the University of Washington.<ref name=":0" /> Between 1949 and 1952, Lanczos worked for the National Bureau of Standards (now the National Institute of Standards and Technology) Institute for Numerical Analysis at the University of California at Los Angeles (UCLA).<ref name=":0" /> There, he participated in the Mathematical Tables Project.<ref name=":4" />
In 1942, Lanczos and Gordon Charles Danielson developed a practical technique in Fourier analysis, now known as the fast Fourier transform (FFT).<ref>Danielson, G. C., and C. Lanczos, "Some improvements in practical Fourier analysis and their application to X-ray scattering from liquids," ''J. Franklin Inst.'' '''233''', 365–380 and 435–452 (1942).</ref><ref name=":4" /> But the significance of his discovery was not appreciated at the time, partly because there were no machines to execute this algorithm,<ref name=":5" /> and today the FFT is credited to J. W. Cooley and John Tukey, who published the Cooley–Tukey algorithm in 1965.<ref name=":1">{{cite journal |author1=Michael T. Heideman |author2=Don H. Johnson |author3=C. Sidney Burrus |date=October 1984 |title=Gauss and the History of the Fast Fourier Transform |journal=IEEE ASSP Magazine |volume=1 |issue=4 |page=14 |doi=10.1109/MASSP.1984.1162257 |bibcode=1984IASSP...1...14H }}</ref> (As a matter of fact, similar claims can be made for several other mathematicians, including Carl Friedrich Gauss.<ref name=":1" />) The FFT was implemented on a digital computer for the first time in 1966.<ref name=":4" />
Working in at the U.S. National Bureau of Standards in the District of Columbia after 1949, Lanczos developed a number of techniques for mathematical calculations using digital computers, such as the Lanczos algorithm for determining the eigenvalues of large Hermitian matrices.<ref name=":5" />
In 1949, Lanczos showed that the Weyl tensor, which plays a fundamental role in general relativity, can be obtained from a tensor potential now called the Lanczos potential.<ref name="Lanczos1949">{{cite journal |last=Lanczos |first=Cornelius |date=July 1, 1949 |title=Lagrangian Multiplier and Riemannian Spaces |journal=Reviews of Modern Physics |publisher=American Physical Society (APS) |volume=21 |issue=3 |pages=497–502 |bibcode=1949RvMP...21..497L |doi=10.1103/revmodphys.21.497 |issn=0034-6861 |doi-access=free}}</ref>
In 1952, Lanczos examined the utility of the Chebyshev polynomials in approximating the solution of linear systems.<ref>{{cite journal |last1=Lanczos |first1=Cornelius |year=1952 |title=Solution of systems of linear equations by minimized iterations |journal=Journal of Research of the National Bureau of Standards |volume=49 |issue=1 |pages=33 |doi=10.6028/jres.049.006 |doi-access=free |bibcode=1952NISTJ..49...33L }}</ref>
During the McCarthy era, Lanczos came under suspicion for possible communist links.<ref name=":2" />{{rp|89}} In 1955, he accepted an invitation from Éamon de Valera, then the Prime Minister of Ireland, and moved to the School of Theoretical Physics at the Dublin Institute for Advanced Studies,<ref name=":4" /> where his colleagues included Schrödinger and John Lighton Synge.<ref name=":4" /><ref>{{cite book|title=The Lanczos Method: Evolution and Application|author=Louis Komzsik|page=79|publisher=SIAM|year=2003}}</ref> Shortly after arriving he gave lectures on numerical methods, such as a new approximation for the gamma function he developed.<ref name=":4" /> He remained there until his death in 1974.<ref>[https://www.dias.ie/2010/07/09/lanczos-c/ Cornelius Lanczos] at Dublin Institute for Advanced Studies</ref> He wrote many scientific papers and books during this period; he also became interested in some newly developed ideas in mathematical physics, notably Schwartz distributions and Sobolev spaces.<ref name=":4" /> Despite being a victim of Joseph McCarthy, Lanczos still referred to the United States as his "dream land" on conversations with those he knew.<ref name=":4" />thumb|A commemorative plaque at the childhood home of Lanczos. The inscription reads: "Kornél Lánczos spent the years of his youth in this house from 1893 to 1911, an excellent practitioner of mathematics and physics, a world-renowned scientist of quantum mechanics and relativity."In 1960, he won the Chauvenet Prize from the Mathematical Association of America (MAA) for a paper explaining how to decompose an arbitrary rectangular matrix into three, the middle of which is diagonal and the other two orthogonal. This technique is now recognized as singular value decomposition, of use in computer science and computational mathematics.<ref name=":4" />
Lanczos resampling is based on a windowed sinc function as a practical upsampling filter approximating the ideal sinc function, now widely used in video up-sampling for digital zoom applications and image scaling. It was invented by Claude Duchon, who named it after Lanczos due to Duchon's use of the sigma approximation in constructing the filter, a technique created by Lanczos.<ref name="duchon">{{cite journal |last1=Claude |first1=Duchon |date=1979-08-01 |title=Lanczos Filtering in One and Two Dimensions |journal=Journal of Applied Meteorology |volume=18 |issue=8 |pages=1016–1022 |bibcode=1979JApMe..18.1016D |doi=10.1175/1520-0450(1979)018<1016:LFIOAT>2.0.CO;2 |doi-access=free}}</ref>
During his career, he was invited to lecture of various topics of mathematical physics at many different institutions.<ref name=":0" /> he maintained contact with his doctoral supervisor Ortvay before the War,<ref name=":4" /> and occasionally returned to Budapest to lecture on various topics, such as the Stark effect (1930) and Hamilton's principle and canonical equations in classical mechanics (1933).<ref name=":3" /> Throughout his life, Lanczos maintained his conviction that mathematics should not be separated from its history; he lectured on this topic with great enthusiasm.<ref name=":5" />
In 1974, he published a paper on the vector potential in curved spacetime.<ref name=":4" /> He died in Budapest that same year of a sudden heart attack during a summer visit. His collected works, six volumes in all, are held at North Carolina State University in collaboration with the Eötvös Physical Society in Budapest.<ref name=":3" /> He was of sound mind mind up until the day he died, when he was working on the Fourier analysis of random sequences, a topic he was scheduled to lecture on in Dublin in July that year.<ref name=":4" />
== Publications == ===Books===
* {{Cite book |title=The Variational Principles of Mechanics |publisher=University of Toronto Press |year=1970 |isbn=0-486-65067-7 |edition=4th |ref=none}} Dedicated to Albert Einstein. This is a graduate text on mechanics.<ref>{{cite journal |author=Lewis, D. C. |year=1951 |title=Review: ''The Variational Principles of Mechanics'', by C. Lanczos |url=https://www.ams.org/journals/bull/1951-57-01/S0002-9904-1951-09462-8/ |journal=Bull. Amer. Math. Soc. |volume=57 |issue=1, Part 1 |pages=88–91 |doi=10.1090/s0002-9904-1951-09462-8 |doi-access=free}}</ref> He published it shortly after moving to Los Angeles.<ref name=":4" /> In the preface of the first edition (1949) it is described as a two-semester graduate course of three hours weekly. The second edition (1962) contains a new chapter on relativistic mechanics and the third (1966) has an appendix on Noether's theorem for cyclic coordinates. In the fourth edition (1970), Lanczos discusses at length continuum mechanics and makes further use of Noether's theorem.<ref>{{Cite journal |last=Jeffreys |first=Bertha |date=1973 |title=The Variational Principles of Mechanics |journal=Mathematical Gazette |volume=57 |issue=399 |pages=81 |doi=10.2307/3615196 |jstor=3615196 }}</ref> * {{Cite book |title=Applied Analysis |publisher=Prentice Hall |year=1956 |ref=none}} Reprinted 2010 by Dover Publications. {{ISBN|978-0-486-65656-4}}. An exposition of his investigations of ideas in the boundary between classical and numerical analysis illustrated by worked examples, topics covered include large scale linear systems, harmonic analysis, data analysis, numerical quadrature and power series expansions.<ref>{{cite journal|author=Todd, John|author-link=John Todd (computer scientist)|title=Review: ''Applied Analysis'', by C. Lanczos|journal=Bull. Amer. Math. Soc.|year=1958|volume=64|issue=4|pages=210–211|url=https://www.ams.org/journals/bull/1958-64-04/S0002-9904-1958-10215-3/|doi=10.1090/s0002-9904-1958-10215-3|doi-access=free}}</ref> The chapter on numerical quadrature was inspired by a number of problems posed by Schrödinger.<ref name=":4" /> * {{Cite book |title=Linear Differential Operators |publisher=Van Nostrand |year=1961 |oclc=1213191}} * {{Cite book |title=Albert Einstein and the Cosmic World Order |publisher=Interscience Publishers |year=1965 |oclc=530604}} Based on six lectures delivered at the University of Michigan in the spring of 1962. * {{Cite book |title=Discourse on Fourier Series |publisher=Oliver & Boyd |year=1966 |location=Edinburgh |oclc=1222573 |ref=none}} This book was written for advanced undergraduates and graduate students in mathematics, physics, and engineering. Lanczos clarifies the meanings of terms such as limit or uniform convergence; and discusses the Sturm–Liouville problems, the Fejér mean, the Gibbs phenomenon, Fourier versus Taylor series, Fourier integrals, the Fourier transform, and the Laplace transform.<ref name=":6">{{Cite journal |last=Jordinson |first=Ralph |date=1968 |title=Review: ''Discourse on Fourier Series'' by Cornelius Lanczos |journal=Proceedings of the Edinburgh Mathematical Society |volume=16 |issue=2 |pages=183–184 |doi=10.1017/S0013091500012645}}</ref><ref name=":8" /> He goes into the historical development of the subject,<ref name=":6" /> but does not cover Lebesgue integration or function spaces.<ref name=":8">{{Cite journal |last=Dickinson |first=David |date=June–July 1967 |title=Review: ''Discourse on Fourier Series'' by Cornelius Lanczos |journal=American Mathematical Monthly |volume=74 |issue=6 |pages=750 |doi=10.2307/2314303 |jstor=2314303 }}</ref> * ''Numbers without End'', Edinburgh: Oliver & Boyd. 1968. * {{Cite book |title=Judaism and Science |publisher=Leeds University Press |year=1970 |isbn=978-0-853-16021-2 |ref=none}} Lectures given in honor of Selig Brodetsky. * {{Cite book |title=Space Through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein |publisher=Academic Press |year=1970 |isbn=9780124358508 |ref=none}} Based on a series of lectures given to mathematicians, physicists, chemists, engineers, and philosophers at North Carolina State University in 1968, Lanczos overviews the history of geometry from the time of the ancient Greeks up until the early twentieth century.<ref name=":9" /><ref name=":10">{{Cite journal |last=Coulson |first=C. A. |date=June 1971 |title=''Space through the Ages'' by Cornelius Lanczos |journal=Mathematical Gazette |volume=55 |issue=393 |pages=334 |doi=10.2307/3615043 |jstor=3615043 }}</ref> He elaborates upon Euclidean geometry, tensor analysis, and the abstract vector spaces of Hilbert and Banach, and briefly describes projective geometry.<ref name=":10" /> He does not, however, discuss topology.<ref name=":9">{{Cite journal |last=Jammer |first=Max |author-link=Max Jammer |date=1970 |title=''Space through the Ages: The Evolution of Geometrical Ideas from Pythagoras to Hilbert and Einstein''. Cornelius Lanczos. Academic Press, New York, 1970. X, 322 Pp., Illus. $11.50. |url=https://www.science.org/doi/10.1126/science.170.3963.1183.a |journal=Science |volume=170 |issue=3963 |pages=1183 |doi=10.1126/science.170.3963.1183.a|url-access=subscription }}</ref> * {{Cite book |title=The Einstein Decade (1905-1915) |publisher=Elek |year=1974 |isbn=978-0-236-17632-8 |location=London |ref=none}} In this book, Lanczos made use of his fluency in the German language as his grasp of to mathematics and physics discuss in detail the scientific publications of Albert Einstein during that time.<ref name=":4" /> * {{Cite book |title=Cornelius Lanczos: Collected Published Papers with Commentaries |publisher=North Carolina State University |year=1998 |isbn=0-929493-01-X |editor-last=Davis |editor-first=William R.}}
===Articles=== * {{cite journal |last=Lanczos |first=Kornel |title=Über eine stationäre Kosmologie im Sinne der Einsteinschen Gravitationstheorie |journal=Zeitschrift für Physik |publisher=Springer Science and Business Media LLC |volume=21 |issue=1 |year=1924 |issn=1434-6001 |doi=10.1007/bf01328251 |pages=73–110 |bibcode=1924ZPhy...21...73L |s2cid=122902359 |language=de}} Translated reprint {{Cite journal |last=Lanczos |first=K. (Kornel) |author-link=Cornelius Lanczos |title=On a Stationary Cosmology in the Sense of Einstein's Theory of Gravitation |journal=General Relativity and Gravitation |volume=29 |pages=363–399 |year=1997 |issue=3 |doi=10.1023/A:1010277120072 |url=https://archive.org/details/zeitschrift-fuer-physik-a-atoms-and-nuclei_1924_21/page/73/mode/1up}} * {{cite journal |last=Lanczos |first=Kornel |url=https://nvlpubs.nist.gov/nistpubs/jres/045/jresv45n4p255_A1b.pdf |title=An iteration method for the solution of the eigenvalue problem of linear differential and integral operators |journal=Journal of Research of the National Bureau of Standards |volume=45 |issue=4 |date=October 1950 |page=255 |location=Los Angeles |doi=10.6028/jres.045.026 }} Research Paper 2133. September 1949.
==See also== {{Portal|Biographies|Mathematics}}
* The conjugate gradient method for solving systems of linear equations
*Hungarian diaspora {{clear}}
== References == {{Reflist}}
== External links == * {{MacTutor Biography|id=Lanczos}} * {{MathGenealogy|id=148153}} * [https://web.archive.org/web/20061205022927/http://physics.ncsu.edu/lanczos/ Cornelius Lanczos, Collected published papers with commentaries], published by North Carolina State University * [http://www.maths.manchester.ac.uk/~higham/photos/lanczos/index.htm Photo gallery of Lanczos] by Nicholas Higham * [http://guettel.com/lanczos/ Series of historic video tapes] produced in 1972, digitalized on the occasion of the 120th anniversary of Cornelius Lanczos's birth * {{cite book|author=Brendan Scaife|author-link=Brendan Scaife|title=Studies in Numerical Analysis: Papers in Honour of Cornelius Lanczos|location=Dublin; London; New York|publisher=Academic Press|year=1974|isbn=0-12-621150-7|url-access=registration|url=https://archive.org/details/studiesinnumeric0000unse}}
{{Chauvenet Prize recipients}}
{{Authority control}}
{{DEFAULTSORT:Lanczos, Cornelius}} Category:1893 births Category:1974 deaths Category:People from Székesfehérvár Category:20th-century Hungarian Jews Category:Hungarian emigrants to the United States Category:20th-century Hungarian mathematicians Category:American expatriates in the Republic of Ireland Category:20th-century Hungarian physicists Category:20th-century Irish mathematicians Category:Numerical analysts Category:American relativity theorists Category:Jewish American physicists Category:Mathematicians from Austria-Hungary Category:Academics of the Dublin Institute for Advanced Studies Category:Fellows of the American Physical Society Category:Victims of McCarthyism Category:Irish relativity theorists Category:National Institute of Standards and Technology people