MA4229 Lectures 5, 6 Week 5 August 24, 27, 2010 Chapter 6 Uniform approximation by polynomials

Weierstrass Theorem Theorem 6.1 For every andthere existswith and K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, 1885 (II).

Monotone Operators Theorem 6.2 If is monotone if Definition A linear operator monotone operators such that is a sequence of uniformly forthenuniformly for Proof pages study Figure 6.1 on page 63

Remarkable Basis for occurs at define polynomials of degree Question Prove that these form a basis for on [0,1] For Question Prove that the maximum value of See Exercise 6.6 on page 70. Question Show that

The Bernstein Operator define the operator by For Question Show that is a linear operator. Question Show that is not a projection. Question Show that mapsonto Question Show thatis monotone. Question Show that whenever

Derivatives of The Bernstein Operator Theorem 6.4 If then Proof pages

Tutorial 3 Due Tuesday 7 September Derive Equation 6.27 on page 66 Pages Exercises 6.1, 6.2, 6.5, 6.9 Derive Equation 6.28 on page 66