Orthogonal polynomials | Polynomials | Approximation theory | Special hypergeometric functions
The Chebyshev polynomials are two sequences of polynomials related to the cosine and sine functions, notated as and . They can be defined in several equivalent ways, one of which starts with trigonometric functions: The Chebyshev polynomials of the first kind are defined by Similarly, the Chebyshev polynomials of the second kind are defined by That these expressions define polynomials in may not be obvious at first sight, but follows by rewriting and using de Moivre's formula or by using the angle sum formulas for and repeatedly. For example, the double angle formulas, which follow directly from the angle sum formulas, may be used to obtain and , which are respectively a polynomial in and a polynomial in multiplied by . Hence and . An important and convenient property of the Tn(x) is that they are orthogonal with respect to the inner product: and Un(x) are orthogonal with respect to another, analogous inner product, given below. The Chebyshev polynomials Tn are polynomials with the largest possible leading coefficient whose absolute value on the interval [−1, 1] is bounded by 1. They are also the "extremal" polynomials for many other properties. Chebyshev polynomials are important in approximation theory because the roots of Tn(x), which are also called Chebyshev nodes, are used as matching points for optimizing polynomial interpolation. The resulting interpolation polynomial minimizes the problem of Runge's phenomenon and provides an approximation that is close to the best polynomial approximation to a continuous function under the maximum norm, also called the "minimax" criterion. This approximation leads directly to the method of Clenshaw–Curtis quadrature. These polynomials were named after Pafnuty Chebyshev. The letter T is used because of the alternative transliterations of the name Chebyshev as Tchebycheff, Tchebyshev (French) or Tschebyschow (German). (Wikipedia).
Chebyshev Polynomials of the First Kind (synthwave enumeration)
This synthwave enumeration shows the first 20 Chebyshev polynomials of the first kind. The polynomials are plotted on the domain [-1,1] (and the range happens to also be [-1,1]). These polynomials are famously related to the Fibonacci numbers (and similar recurrence sequences) as well as t
From playlist Synthwave Mathematics
Chebyshev Polynomials via cos(1°)
In this video, we introduce and motivate the Chebyshev polynomials (1st kind) in proving that the cosines of numerous angles must be irrational numbers. No advanced math beyond high school trigonometry is needed to understand this video, which is quite remarkable considering the many real-
From playlist Math
Quintic Equation From Chebyshev Polynomial
Quintic Equation From Chebyshev Polynomial. We will see an introduction to Chebyshev Polynomial, which is a polynomial in terms of cosine. And then we will use this polynomial to create a solvable quintic equation. This kind of polynomial equation from trigonometric identities is very clas
From playlist Trigonometry, but for fun!
Advice for Research Mathematicians | Rational Trigonometry and Spread Polynomials II | Wild Egg Math
Spread polynomials arise in Rational Trigonometry as variants of the Chebyshev polynomials of the first kind. However the spread polynomials arise in a purely algebraic setting, without any need for appeal to "transcendental functions" which can't actually be evaluated -- such as cos x or
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Algebra - Ch. 5: Polynomials (1 of 32) What is a Polynomial?
Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is a polynomial. An algebraic expression with 2 or more terms. I will also explain what is a monomial, binomial, trinomial, and polynomial of 4 terms; terms, and factors. To donate: http
From playlist ALGEBRA CH 5 POLYNOMIALS
ch2 A: Chebyshev nodes. Wen Shen
Wen Shen, Penn State University. Lectures are based on my book: "An Introduction to Numerical Computation", published by World Scientific, 2016. See promo video: https://youtu.be/MgS33HcgA_I
From playlist CMPSC/MATH 451 Videos. Wen Shen, Penn State University
Advice for Maths Exploration | Chebyshev and Spread Polynumbers: the remarkable Goh factorization
A key challenge for amateur mathematicians is finding a fruitful and accessible and interesting area for investigation. This is not so easy: classical number theory is certainly very interesting but it is highly difficult, perhaps even unrealistic, to hope to make really new discoveries he
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Continuous-Time Chebyshev and Elliptic Filters
http://AllSignalProcessing.com for more great signal processing content, including concept/screenshot files, quizzes, MATLAB and data files. An introduction to the characteristics and definition of analog Chebyshev types I and II and elliptic filters.
From playlist Infinite Impulse Response Filter Design
How to classify and determine lc degree of a polynomial
👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1
From playlist Classify Polynomials | Equations
How to reorder and classify a polynomial based on it's degree and number of terms
👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1
From playlist Classify Polynomials | Equations
More bases of polynomial spaces | Wild Linear Algebra A 21 | NJ Wildberger
Polynomial spaces are excellent examples of linear spaces. For example, the space of polynomials of degree three or less forms a linear or vector space which we call P^3. In this lecture we look at some more interesting bases of this space: the Lagrange, Chebyshev, Bernstein and Spread po
From playlist WildLinAlg: A geometric course in Linear Algebra
Advice for research mathematicians | The joy of maxel number theory: Chebyshev polys I | Wild Egg
We are advocating a larger view of number theory which goes from arithmetic with numbers to polynumbers to maxels. In this lecture we have a look at the Chebyshev polynumbers of the first kind from this larger linear algebraic point of view. Some surprises are in store! ******************
From playlist Maxel inverses and orthogonal polynomials (non-Members)
Advice for Research Mathematicians | The joy of maxel number theory: Chebyshev Polys 2 | Wild Egg
We extend our newish approach to families of orthogonal polynomials / polynumbers involving creating two dimensional arrays, or maxels, from them to the case of the Chebyshev polynomials of the second kind. These are very important in representation theory of Lie groups and Lie algebras,
From playlist Maxel inverses and orthogonal polynomials (non-Members)
A Tour of Skein Modules by Rhea Palak Bakshi
PROGRAM KNOTS THROUGH WEB (ONLINE) ORGANIZERS: Rama Mishra, Madeti Prabhakar, and Mahender Singh DATE & TIME: 24 August 2020 to 28 August 2020 VENUE: Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through onl
From playlist Knots Through Web (Online)
Mioara Joldes: Validated symbolic-numerci algorithms and practical applications in aerospace
In various fields, ranging from aerospace engineering or robotics to computer-assisted mathematical proofs, fast and precise computations are essential. Validated (sometimes called rigorous as well) computing is a relatively recent field, developed in the last 20 years, which uses numerica
From playlist Probability and Statistics
Classify a polynomial and determine degree and leading coefficient
👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1
From playlist Classify Polynomials | Equations
Classify a polynomial and determine degree and leading coefficient
👉 Learn how to classify polynomials. A polynomial is an expression of the sums/differences of two or more terms having different integer exponents of the same variable. A polynomial can be classified in two ways: by the number of terms and by its degree. A monomial is an expression of 1
From playlist Classify Polynomials | Equations
Mod-01 Lec-05 Error in the Interpolating polynomial
Elementary Numerical Analysis by Prof. Rekha P. Kulkarni,Department of Mathematics,IIT Bombay.For more details on NPTEL visit http://nptel.ac.in
From playlist NPTEL: Elementary Numerical Analysis | CosmoLearning Mathematics