Tamas Erdelyi (mathematician)
Encyclopedia
Tamás Erdélyi is a Hungarian
born mathematician
, currently
working at Texas A&M University
. His main areas of research are related to polynomial
s and their approximations, although he also works in other areas of applied mathematics
.
, Hungary. From 1980 to 1985 he studied mathematics at the ELTE
in Budapest, where he received his diploma. After graduation, he worked for two years as a research assistant at the Mathematics Institute of the Hungarian Academy of Sciences
. He later pursued his graduate studies at the University of South Carolina
(1987-88) and the at the Ohio State University
(1988-89). He received his Ph.D. from the University of
South Carolina in 1989. He was a postdoctoral fellow at Ohio State University (1989-92), Dalhousie University
(1992-93), Simon Fraser University
(1993-95), and finally at the University of Copenhagen
(1996-97). In 1995, he began working at the Texas A&M University in
College Satation Texas, where he is currently a professor of mathematics.
represents one of his most cited papers.
In 1995, he finished his Springer-Verlag graduate text Polynomials and Polynomial Inequalities, co-authored with Peter Borwein
, and including an appendix proving the irrationality of zeta(2) and zeta(3). Later that year he showed that Müntz's theorem holds on every compact
subset of the positive real axis of the Lebesgue measure
. His bounded Remez-type inequality for Müntz polynomials in the non-dense case also allowed him to resolve Newman's Product Problem. In the same year he also proved a Bernstein's inequality for exponential sum
s, the subject of an earlier conjecture by G.G. Lorentz.
Erdelyi has also published papers dealing with other important inequalities for exponential sums and linear combinations of shifted Gaussian
s. Early in the twenty first century he proved two of Saffari's conjectures, the Phase Problem
and the Near Orthogonality Conjecture. In 2007, working with Borwein, Ferguson, and Lockhart, he settled Littlewood's Problem 22. He is an expert on ultraflat and flat sequences of unimodular polynomials, having published papers on the location of zeros for polynomials with constrained coefficients, and on orthogonal polynomials
. He has also made significant contributions to the Integer Chebyshev Problem, worked with Harvey Friedman
on recursion theory
, and, together with Borwein, disproved a conjecture made by the Chudnovsky brothers
.
Erdelyi's more recent work has focussed on problems in the interface of harmonic analysis
and number theory
, and the Mahler measure of constrained polynomials. He contributed substantially to Chowla's Cosine Problem by proving Bourgain and Ruzsa type results for the maximum and minimum of Littlewood cosine polynomials. One of his Bernstein type inequalities for rational function
s is now referred to as the Borwein-Erdelyi inequality. He is also known for establishing the Full Müntz Theorem with Borwein and Johnson, and has some partial results related to questions raised by Paul Erdős
.
Hungary
Hungary , officially the Republic of Hungary , is a landlocked country in Central Europe. It is situated in the Carpathian Basin and is bordered by Slovakia to the north, Ukraine and Romania to the east, Serbia and Croatia to the south, Slovenia to the southwest and Austria to the west. The...
born mathematician
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, currently
working at Texas A&M University
Texas A&M University
Texas A&M University is a coeducational public research university located in College Station, Texas . It is the flagship institution of the Texas A&M University System. The sixth-largest university in the United States, A&M's enrollment for Fall 2011 was over 50,000 for the first time in school...
. His main areas of research are related to polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s and their approximations, although he also works in other areas of applied mathematics
Applied mathematics
Applied mathematics is a branch of mathematics that concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge...
.
Life, education and positions
Tamas Erdelyi was born on September 13, 1961, in BudapestBudapest
Budapest is the capital of Hungary. As the largest city of Hungary, it is the country's principal political, cultural, commercial, industrial, and transportation centre. In 2011, Budapest had 1,733,685 inhabitants, down from its 1989 peak of 2,113,645 due to suburbanization. The Budapest Commuter...
, Hungary. From 1980 to 1985 he studied mathematics at the ELTE
Elte
Elte can refer to the following:* Elte, North Rhine-Westphalia, former township now integrated into Rheine, Germany.* Eötvös Loránd University, a University in Budapest.* Harry Elte, Dutch architect.* E. L. Elte, Dutch mathematician...
in Budapest, where he received his diploma. After graduation, he worked for two years as a research assistant at the Mathematics Institute of the Hungarian Academy of Sciences
Hungarian Academy of Sciences
The Hungarian Academy of Sciences is the most important and prestigious learned society of Hungary. Its seat is at the bank of the Danube in Budapest.-History:...
. He later pursued his graduate studies at the University of South Carolina
University of South Carolina
The University of South Carolina is a public, co-educational research university located in Columbia, South Carolina, United States, with 7 surrounding satellite campuses. Its historic campus covers over in downtown Columbia not far from the South Carolina State House...
(1987-88) and the at the Ohio State University
Ohio State University
The Ohio State University, commonly referred to as Ohio State, is a public research university located in Columbus, Ohio. It was originally founded in 1870 as a land-grant university and is currently the third largest university campus in the United States...
(1988-89). He received his Ph.D. from the University of
South Carolina in 1989. He was a postdoctoral fellow at Ohio State University (1989-92), Dalhousie University
Dalhousie University
Dalhousie University is a public research university located in Halifax, Nova Scotia, Canada. The university comprises eleven faculties including Schulich School of Law and Dalhousie University Faculty of Medicine. It also includes the faculties of architecture, planning and engineering located at...
(1992-93), Simon Fraser University
Simon Fraser University
Simon Fraser University is a Canadian public research university in British Columbia with its main campus on Burnaby Mountain in Burnaby, and satellite campuses in Vancouver and Surrey. The main campus in Burnaby, located from downtown Vancouver, was established in 1965 and has more than 34,000...
(1993-95), and finally at the University of Copenhagen
University of Copenhagen
The University of Copenhagen is the oldest and largest university and research institution in Denmark. Founded in 1479, it has more than 37,000 students, the majority of whom are female , and more than 7,000 employees. The university has several campuses located in and around Copenhagen, with the...
(1996-97). In 1995, he began working at the Texas A&M University in
College Satation Texas, where he is currently a professor of mathematics.
Works
Erdelyi started his career studying Markov and Bernstein inequalities for constrained polynomials in the late eighties. In his Ph.D. dissertation he extended many important polynomial inequalities to generalized polynomials by writing the generalized degree in place of the ordinary. His trigonometric work on Remez inequalityRemez inequality
In mathematics, the Remez inequality, discovered by the Ukrainian mathematician Evgeny Yakovlevich Remez , gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.-The inequality:...
represents one of his most cited papers.
In 1995, he finished his Springer-Verlag graduate text Polynomials and Polynomial Inequalities, co-authored with Peter Borwein
Peter Borwein
Peter Benjamin Borwein is a Canadian mathematicianand a professor at Simon Fraser University. He is known as a co-discoverer of the Bailey-Borwein-Plouffe algorithm for computing π.-First interest in mathematics:...
, and including an appendix proving the irrationality of zeta(2) and zeta(3). Later that year he showed that Müntz's theorem holds on every compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
subset of the positive real axis of the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
. His bounded Remez-type inequality for Müntz polynomials in the non-dense case also allowed him to resolve Newman's Product Problem. In the same year he also proved a Bernstein's inequality for exponential sum
Exponential sum
In mathematics, an exponential sum may be a finite Fourier series , or other finite sum formed using the exponential function, usually expressed by means of the functione = \exp.\,...
s, the subject of an earlier conjecture by G.G. Lorentz.
Erdelyi has also published papers dealing with other important inequalities for exponential sums and linear combinations of shifted Gaussian
GAUSSIAN
Gaussian is a computational chemistry software program initially released in 1970 by John Pople and his research group at Carnegie-Mellon University as Gaussian 70. It has been continuously updated since then...
s. Early in the twenty first century he proved two of Saffari's conjectures, the Phase Problem
Phase problem
In physics the phase problem is the name given to the problem of loss of information concerning the phase that can occur when making a physical measurement. The name itself comes from the field of x-ray crystallography, where the phase problem has to be solved for the determination of a structure...
and the Near Orthogonality Conjecture. In 2007, working with Borwein, Ferguson, and Lockhart, he settled Littlewood's Problem 22. He is an expert on ultraflat and flat sequences of unimodular polynomials, having published papers on the location of zeros for polynomials with constrained coefficients, and on orthogonal polynomials
Orthogonal polynomials
In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical polynomials, the Chebyshev polynomials, and the...
. He has also made significant contributions to the Integer Chebyshev Problem, worked with Harvey Friedman
Harvey Friedman
Harvey Friedman is a mathematical logician at Ohio State University in Columbus, Ohio. He is noted especially for his work on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary...
on recursion theory
Recursion theory
Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...
, and, together with Borwein, disproved a conjecture made by the Chudnovsky brothers
Chudnovsky brothers
The Chudnovsky brothers are American mathematicians known for their wide mathematical ability, their home-built supercomputers, and their close working relationship....
.
Erdelyi's more recent work has focussed on problems in the interface of harmonic analysis
Harmonic analysis
Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves. It investigates and generalizes the notions of Fourier series and Fourier transforms...
and number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, and the Mahler measure of constrained polynomials. He contributed substantially to Chowla's Cosine Problem by proving Bourgain and Ruzsa type results for the maximum and minimum of Littlewood cosine polynomials. One of his Bernstein type inequalities for rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
s is now referred to as the Borwein-Erdelyi inequality. He is also known for establishing the Full Müntz Theorem with Borwein and Johnson, and has some partial results related to questions raised by Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
.