Chebyshev polynomials—From approximation theory to algebra and number theory

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.

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Identities connecting the Chebyshev polynomials

The Mathematical Gazette ◽

10.1017/mag.2016.110 ◽

2016 ◽

Vol 100 (549) ◽

pp. 450-459 ◽

Cited By ~ 1

Author(s):

Jonny Griffiths

Keyword(s):

Differential Equations ◽

Weight Function ◽

Orthogonal Polynomial ◽

Recurrence Relation ◽

Approximation Theory ◽

Chebyshev Polynomials ◽

Pell Equation ◽

Fibonacci Polynomials ◽

The Third ◽

Polynomial Family

There are many families of polynomials in mathematics, and they often occur naturally in pairs. The Fibonacci polynomials and the Lucas polynomials, for example, are generated by the same recurrence relation but with different starting values, and there are many identities that link the two families [1]. The same is true for the Chebyshev polynomials of the first and second kinds, Tn (x) and Un (x) [2], respectively. There are two further polynomial families that are less well-known, the Chebyshev polynomials of the third and fourth kinds, Vn (x) and Wn (x) [3], respectively. Each of the four kinds is an example of an orthogonal polynomial family Pn (x), where for some appropriate weight function W (x), whenever n ≠ m. The families Tn (x) and Un (x) in particular are ubiquitous in their mathematical uses, in approximation theory, in differential equations, and in solving the Pell equation, to name but three. There are also many connections between Tn (x), Un (x), Vn (x) and Wn (x), some of which are explored here, and some of which we hope are new.

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Error-correcting codes and Minkowski’s conjecture

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-010-0004-y ◽

2010 ◽

Vol 45 (1) ◽

pp. 37-49 ◽

Cited By ~ 2

Author(s):

Peter Horak

Keyword(s):

Number Theory ◽

Group Theory ◽

Linear Algebra ◽

Approximation Theory ◽

Error Correcting Codes ◽

The Other ◽

Theory Approximation

ABSTRACT The goal of this paper is twofold. The main one is to survey the latest results on the perfect and quasi-perfect Lee error correcting codes. The other goal is to show that the area of Lee error correcting codes, like many ideas in mathematics, can trace its roots to the Phytagorean theorem a2+b2 = c2. Thus to show that the area of the perfect Lee error correcting codes is an integral part of mathematics. It turns out that Minkowski’s conjecture, which is an interface of number theory, approximation theory, geometry, linear algebra, and group theory is one of the milestones on the route to Lee codes.

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On linear combinations of Chebyshev polynomials

Publications de l Institut Mathematique ◽

10.2298/pim150220001s ◽

2015 ◽

Vol 97 (111) ◽

pp. 57-67

Author(s):

Dragan Stankov

Keyword(s):

Number Theory ◽

Chebyshev Polynomials ◽

Infinite Sequence ◽

Real Numbers ◽

Linear Combinations ◽

Salem Numbers ◽

Limit Points

We investigate an infinite sequence of polynomials of the form: a0Tn(x) + a1Tn?1(x) + ... + amTn?m(x) where (a0, a1,..., am) is a fixed m-tuple of real numbers, a0, am ? 0, Ti(x) are Chebyshev polynomials of the first kind, n = m, m+ 1, m+ 2,.... Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An application in number theory, more precise, in the theory of Pisot and Salem numbers, is presented.

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DETERMINACY AND INDETERMINACY OF GAMES PLAYED ON COMPLETE METRIC SPACES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714000069 ◽

2014 ◽

Vol 90 (2) ◽

pp. 339-351

Author(s):

LIOR FISHMAN ◽

TUE LY ◽

DAVID SIMMONS

Keyword(s):

Number Theory ◽

Approximation Theory ◽

Diophantine Approximation ◽

Metric Spaces ◽

Complete Metric Spaces ◽

Large Audience ◽

Schmidt’S Game

AbstractSchmidt’s game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory and dynamics. Recently, many new results have been proven using this game. In this paper we address determinacy and indeterminacy questions regarding Schmidt’s game and its variations, as well as more general games played on complete metric spaces (for example, fractals). We show that, except for certain exceptional cases, these games are undetermined on certain sets. Judging by the vast numbers of papers utilising these games, we believe that the results in this paper will be of interest to a large audience of number theorists as well as set theorists and logicians.

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