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set9 - $ Set 9 Minimax(Best and Near-minimax Polynomial...

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a39 a38 a36 a37 Set 9: Minimax (Best) and Near-minimax Polynomial Approximation Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1
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a39 a38 a36 a37 Approximation Outline Best (Minimax) Polynomial Approximation – 10.8 Chebyshev (Near Minimax) Approximation – 10.8 Generalized Fourier Series – 10.1 Orthogonal Polynomials – 10.1 Least Squares approximation – 10.1,10.7 and notes Chebyshev Economization – 10.8 and notes Discrete Least Squares approximation – 10.7 and notes 2
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a39 a38 a36 a37 Best Polynomial Approximation Let P n be the space of polynomials of degree at most n . Problem 9.1. min p n P n bardbl f p n bardbl p * n = argmin bardbl f p n bardbl denotes the minimizer E * n ( f ) = bardbl f p * n bardbl denotes the minimal error Note. Bounds in this norm guarantee that the pointwise error does not exceed a particular amount. 3
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a39 a38 a36 a37 Lower Bound Characterization Theorem 9.1. (De La Vall ´ ee-Poussin) Let p n ( x ) be a polynomial of degree n with deviations from f ( x ) on [ a, b ] f ( x j ) p n ( x j ) = ( 1) j ǫ j , 0 j n + 1 a x 0 < x 1 < · · · < x n < x n +1 b either j ǫ j > 0 or j ǫ j < 0 . The error E * n ( f ) is bounded below by min j | ǫ j | ≤ E * n ( f ) 4
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a39 a38 a36 a37 Lower Bound Characterization Proof. (Isaacson and Keller) Suppose ˜ p n ( x ) has degree n and is such that bardbl f ˜ p n bardbl < min j | ǫ j | = μ Consider the polynomial of degree n , ˜ p n p n at x j , 0 j n + 1 ˜ p ( x j ) p n ( x j ) = ( f ( x j ) p n ( x j )) ( f ( x j ) ˜ p n ( x j )) = ( 1) j ǫ j ( f ( x j ) ˜ p n ( x j )) by assumption | f ( x j ) ˜ p n ( x j ) | < μ ≤ | ǫ j | sign p ( x j ) p n ( x j )) = sign ( f ( x j ) p n ( x j )) n + 2 points of alternating sign for ˜ p ( x ) p n ( x ) implies n + 1 roots and ˜ p n ( x ) p n ( x ) which is a contradiction. 5
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a39 a38 a36 a37 Optimal Characterization Theorem 9.2. (Chebyshev) A polynomial of degree at most n , p * n ( x ) is an optimal approximation of f ( x ) on [ a, b ] with respect to bardbl f p n bardbl if and only if f ( x ) p * n ( x ) = ± E * n ( f ) , with alternating sign changes, at least n + 2 times in [ a, b ] . The polynomial p * n ( x ) is unique.
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