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

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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). http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem

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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 62-65. study Figure 6.1 on page 63

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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

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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

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Derivatives of The Bernstein Operator Theorem 6.4 If then Proof pages 68-69. http://en.wikipedia.org/wiki/File:Bernstein_Approximation.gif

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Tutorial 3 Due Tuesday 7 September Derive Equation 6.27 on page 66 Pages 69-71 Exercises 6.1, 6.2, 6.5, 6.9 Derive Equation 6.28 on page 66

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