MA4229 Lectures 9, 10 Weeks 5-7 Sept 7 - Oct 1, 2010 Chapter 7 The theory of minimax approximation Chapter 8 The exchange algorithm

Minimax Approximation In this lecture continuous functions on the interval denotes the set of REAL Question Givencompute that minimizes Question What is the relationship between and

Error Reduction is nonempty and closed, If Question Explain why. then are continuous,

Tutorial 4 Due Tuesday 14 September Pages Exercises 7.1, 7.4, 7.5, 7.6, 7.7, 7.8

Conditions for Minimization Theorem 7.1 (page 75) Let Then and subspace, be a be closed and and ( is a best approximation fromto ) iff there does not existthat satisfies

The Haar Condition (pages 76-77) Consider the following four conditions for an (n+1)-dimensional subspace 1. If a function in A has > n roots it equals zero 2. Given k < (n+1) distinct points in [a,b] there exists a function in A whose roots are exactly these points and that changes sign at each of these points. In particular there exists an element in A that has no roots. 3. If a nonzero function in A has j roots in [a,b] and k of these are in (a,b) and the function does not change sign then (j+k) < (n+1). 4. All (n+1)x(n+1) matrices of values of functions in a basis for A at (n+1) distinct points in [a,b] are nonsingular Then 1, 3, 4 are equivalent and imply 2.

The Characterization Condition Theorem 7.2 (pages 77-78) If is an (n+1)-dimensional Haar subspace and is a best approximation from A to f whenthen Question State the exact conditions Theorem 7.3 (page 78) describes the relationship between Chebyshev polynomials and best approximation. Explain this relationship Theorem 7.4 (page 79) describes necessary and sufficient conditions for best approximation (from a Haar subpace of dimension (n+1) ) over a reference set of (n+2) points. Explain this relationship.

Uniqueness Theorem 7.5 (pages 77-78) If satisfies is a reference and Then r has at least (n+1) zeros in [a,b] if every double zero is counted twice. Theorem 7.6 There is exactly one best approximation. Explain the conditions for this to hold and give an outline of the proof.

Important Inequalities Theorem 7.7 (pages 77-78) If is such that then is a Haar subspace with dim(A) = (n+1) satisfies and Proof pages (note how the proofs of both Thm 7.6 and Thm 7.6 use Thm 7.6)

Tutorial 5 Due Tuesday 5 October 1. Do Exercise 7.10 on page Read sections 8.1, 8.2 and describe all ways of building a new reference from Figure 8.1 that satisfies Equations (8.12) and (8.13). 2. Compute the best approximation from to over the set a. The two equations in (7.25) on page 79, and using: b. The three equations in (7.27) on page 79 where h is computed as well as the two coefficients of the best approximating polynomial 4. Use Thm 7.7 to explain why 8.12 and 8.13 imply Do Exercises 8.1 and 8.2 on pages

Supplement: Laurent Polynomials Definition An algebraic; Laurent; trigonometric polynomial of deg d Question Prove : an alg. pol. is real valued iff all it coef. are. Question Prove : every trig pol.whereis a Laurent pol., compute the relationship between their coef., and determine conditions forto be real valued.

Supplement: Laurent Polynomials Definition The circle group is the quotient group Question Show a trig. poly. P gives a function Question Show that the trig. poly. has three zeros when it Question How many zeros does the trig. poly. is considered as a function on where is the additive group of real numbers andis the additive group of integers and is the additive group of integer multiples ofThis group has the obvious topology. but only two zeros when it is considered as a function on have considered as a function on the interval, on the circle group? We will identify with the interval

Supplement: Laurent Polynomials Theorem Given two distinct points such that is uniquely determined if Then and let there exists a real valued ) (Show this is the case iff trigonometric polynomial with degree 1, whose set of roots is ) be the Laurent polynomial Proof Let (Show that such thathas roots henceso is real valued iff which happens iff

Supplement: Exercise 7.4 If can have at most 2n roots in any Question Show it is NOT a Haar subspace of So the set of trig. poly. of degree at most 2n is a Haar subspace of would have there exists a Laurent polynomial is a trig. polynomial with degree n, then with degree n such that since otherwise Any such interval more than 2n roots in the set (real valued continuous functions). Hence Condition (1) is Section 7.3 fails. But condition (2) holds by the previous Theorem so these conditions are not equivalent.