Explore advanced concepts in polynomial spaces through this comprehensive lecture on linear algebra. Delve into various interesting bases of the P^3 space, including Lagrange, Chebyshev, Bernstein, and Spread polynomial bases. Learn about Lagrange interpolation polynomials, uniform approximation using Bernstein polynomials, and the recursive definition of Chebyshev polynomials. Discover the relationship between Spread and Chebyshev polynomials, and understand their applications in approximation and integration theories. Gain insights into change of basis matrices and practical exercises to reinforce your understanding of these advanced topics in linear algebra.
Overview
Syllabus
CONTENT SUMMARY: pg 1: @00:08 Introduction review; polynomials of degree 3; Lagrange, Chebyshev, Bernstein, spread polynomials; basis: standard/power, factorial, Taylor; Lagrange polynomials developed ;
pg 2: @ Lagrange development continued; evaluation mapping;
pg 3: @ Lagrange development continued; polynomials that map to the standard basis vectors e1,e2,e3,34 Lagrange interpolation polynomials;
pg 4: @ Lagrange basis; Polynomial that goes through four desired points;
pg 5: @ Uniform approximation and Bernstein polynomials
pg 6: @ reference to Pascal's triangle; Bernstein polynomials named; Bernstein basis;
pg 7: @ view of Bernstein polynomials;
pg 8: @ Show that Bernstein polynomials of a certain degree do form a basis for that corresponding polynomial space; Pascal's triangle; Unnormalized Bernstein polynomials; WLA21_pg8_theorem Bernstein polynomial basis;
pg 9: @ How Bernstein polynomials are used to approximate a given continuous function on an interval;
pg 10: @ Chebyshev polynomials; using a recursive definition; Chebyshev polynomial diagram;
pg 14: @36:56 Spread polynomials relation to Chebyshevs; Spread polynomials advantage over Chebyshev; Pascal's array; Spread polynomials as a source of study @;
pg 15: @39:43 Spread basis; change of basis matrices; moral @ ;
pg 16: @ exercises 21.1-4 ;
pg 17: @43:44 exercises 21.5-7 ; closing remarks @ THANKS to EmptySpaceEnterprise
Introduction
Lagrange polynomials
Lagrange basis
Uniform approximation and Bernstein polynomials
How are Bernstein polynomials used?
Chebyshev polynomials
Chebyshev basis
Chebyshev polynomials in approximation and integration theories
Relation between Spread and Chebyshev polys
Spread basis
Taught by
Insights into Mathematics
