lecture2 - Chapter 2 Function Approximation We saw in the introductory chapter that one key step in the construction of a numerical method to

lecture2 - Chapter 2 Function Approximation We saw in the...

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Chapter 2 Function Approximation We saw in the introductory chapter that one key step in the construction of a numerical method to approximate a definite integral is the approximation of the integrand by a simpler function, which we can integrate exactly. The problem of function approximation is central to many numerical methods: given a continuous function f in an interval [ a, b ], we would like to find a good approximation to it by simpler functions, such as polynomials, trigonometric polynomials, wavelets, rational functions, etc. We are going to measure the accuracy of an approximation using norms and ask whether or not there is a best approximation out of functions from a given family of simpler functions. These are the main topics of this introductory chapter to Approximation Theory. 2.1 Norms A norm on a vector space V over a field K = R (or C ) is a mapping k · k : V ! [0 , 1 ) , which satisfy the following properties: (i) k x k ≥ 0 8 x 2 V and k x k = 0 i x = 0 . (ii) k x + y k  k x k + k y k 8 x, y 2 V . (iii) k λ x k = | λ | k x k 8 x 2 V, λ 2 K . 17
18 CHAPTER 2. FUNCTION APPROXIMATION If we relax (i) to just k x k ≥ 0, we obtain a semi-norm . We recall first some of the most important examples of norms in the finite dimensional case V = R n (or V = C n ): k x k 1 = | x 1 | + . . . + | x n | , (2.1) k x k 2 = p | x 1 | 2 + . . . + | x n | 2 , (2.2) k x k 1 = max {| x 1 | , . . . , | x n |} . (2.3) These are all special cases of the l p norm: k x k p = ( | x 1 | p + . . . + | x n | p ) 1 /p , 1 p  1 . (2.4) If we have weights w i > 0 for i = 1 , . . . , n we can also define a weighted p norm by k x k w,p = ( w 1 | x 1 | p + . . . + w n | x n | p ) 1 /p , 1 p  1 . (2.5) All norms in a finite dimensional space V are equivalent, in the sense that there are two constants c and C such that k x k C k x k β , (2.6) k x k β c k x k , (2.7) for all x 2 V and for any two norms k · k and k · k β defined in V . If V is a space of functions defined on a interval [ a, b ], for example C [ a, b ], the corresponding norms to (2.1)-(2.4) are given by k u k 1 = Z b a | u ( x ) | dx, (2.8) k u k 2 = ✓Z b a | u ( x ) | 2 dx 1 / 2 , (2.9) k u k 1 = sup x 2 [ a,b ] | u ( x ) | , (2.10) k u k p = ✓Z b a | u ( x ) | p dx 1 /p , 1 p  1 (2.11) and are called the L 1 , L 2 , L 1 , and L p norms, respectively. Similarly to (2.5) we can defined a weighted L p norm by k u k p = ✓Z b a w ( x ) | u ( x ) | p dx 1 /p , 1 p  1 , (2.12)
2.2. UNIFORM POLYNOMIAL APPROXIMATION 19 where w is a given positive weight function defined in [ a, b ]. If w ( x ) 0, we get a semi-norm. Lemma 1. Let k · k be a norm on a vector space V then | k x k - k y k |  k x - y k . (2.13) This lemma implies that a norm is a continuous function (on V to R ). Proof. k x k = k x - y + y k  k x - y k + k y k which gives that k x k - k y k  k x - y k . (2.14) By reversing the roles of x and y we also get k y k - k x k  k x - y k . (2.15) 2.2 Uniform Polynomial Approximation There is a fundamental result in approximation theory, which states that any continuous function can be approximated uniformly, i.e. using the norm k · k 1 , with arbitrary accuracy by a polynomial. This is the celebrated Weier- strass Approximation Theorem. We are going to present a constructive proof due to Sergei Bernstein, which uses a class of polynomials that have found widespread applications in computer graphics and animation. Historically,

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