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 polynomial basis. The last comes from Rational Trigonometry.
This is one of a series on Linear Algebra given by N J Wildberger of UNSW.
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 02:18;
pg 2: @03:53 Lagrange developement continued; evaluation mapping;
pg 3: @06:13 Lagrange developement continued; polynomials that map to the standard basis vectors e1,e2,e3,34 (Lagrange interpolation polynomials);
pg 4: @09:12 Lagrange basis; Polynomial that goes through four desired points;
pg 5: @11:39 Uniform approximation and Bernstein polynomials
pg 6: @13:42 reference to Pascal’s triangle; Bernstein polynomials (named); Bernstein basis;
pg 7: @16:37 view of Bernstein polynomials;
pg 8: @18:06 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: @21:04 How Bernstein polynomials are used to approximate a given continuous function on an interval;
pg 10: @24:13 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 @39:17;
pg 15: @39:43 Spread basis; change of basis matrices; moral @42:15 ;
pg 16: @42:40 exercises 21.1-4 ;
pg 17: @43:44 exercises 21.5-7 ; closing remarks @44:38 (THANKS to EmptySpaceEnterprise)
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