2_Interpolation_week3-4 - Interpolation Hector D Ceniceros 1 Approximation Theory Given f C[a b we would like to find a good approximation to it by

# 2_Interpolation_week3-4 - Interpolation Hector D Ceniceros...

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Interpolation * Hector D. Ceniceros 1 Approximation Theory Given f C [ a, b ], we would like to find a “good” approximation to it by “simpler functions”, i.e. functions in a given class (or family) Φ. For example,Φ = P n = { all polynomials of degree n } . A natural problem is that of finding the best approximation to f by func- tions in Φ. But how do we measure the accuracy of any approximation? that is, what norm 1 do we use? We have several choices for norms of functions. The most commonly used are: 1. The max or infinity norm: k f k = sup x [ a,b ] | f ( x ) | . 2. The 2-norm: k f k 2 = ( R b a f 2 ( x ) dx ) 1 2 . 3. The p-norm: k f k p = ( R b a f p ( x ) dx ) 1 p . Later, we will need to consider weighted norms: for some positive function ω ( x ) in [ a, b ] (it could be zero on a finite number of points) we define k f k ω, 2 = Z b a ω ( x ) f 2 ( x ) dx 1 2 . (1) * These are lecture notes for Math 104 A. These notes and all course materials are protected by United States Federal Copyright Law, the California Civil Code. The UC Policy 102.23 expressly prohibits students (and all other persons) from recording lectures or discussions and from distributing or selling lectures notes and all other course materials without the prior written permission of the instructor. 1 A norm k · k is a real valued function on a vector space such that (1) k f k > 0 , f 6≡ 0, (2) k λf k = | λ |k f k for any λ scalar, and (3) k f + g k ≤ k f k + k g k . 1
Then, by best approximation in Φ we mean a function p Φ such that k f - p k ≤ k f - q k , q Φ . Computationally, it is often more efficient to seek not the best approximation but one that is sufficiently accurate and fast converging to f . The central building block for this approximation is the problem of interpolation. 2 Interpolation Let us focus on the case of approximating a given function by a polynomial of degree at most n . Then the interpolation problem can be stated as follows: Given n +1 distinct points, x 0 , x 1 , ..., x n called nodes and corresponding values f ( x 0 ) , f ( x 1 ) , ..., f ( x n ), find a polynomial of degree at most n , P n ( x ), which satisfies (the interpolation property) P n ( x 0 ) = f ( x 0 ) P n ( x 1 ) = f ( x 1 ) . . . P n ( x n ) = f ( x n ) . Let us represent such polynomial as P n ( x ) = a 0 + a 1 x + · · · + a n x n . Then, the interpolation property means P n ( x 0 ) = f ( x 0 ) , P n ( x 1 ) = f ( x 1 ) , · · · , P n ( x n ) = f ( x n ) , which implies a 0 + a 1 x 0 + · · · + a n x n 0 = f ( x 0 ) a 0 + a 1 x 1 + · · · + a n x n 1 = f ( x 1 ) . . . a 0 + a 1 x n + · · · + a n x n n = f ( x n ) . This is a linear system of n +1 equations in n +1 unknowns (the polynomial coefficients a 0 , a 1 , . . . , a n ). In matrix form: 1 x 0 x 2 0 · · · x n 0 1 x 1 x 2 1 · · · x n 1 . . . 1 x n x 2 n · · · x n n a 0 a 1 . . . a n = f ( x 0 ) f ( x 1 ) . . . f ( x n ) (2) 2
Does this linear system have a solution? Is this solution unique? The answer is yes to both. Here is a simple proof. Take f 0, then P n ( x j ) = 0,for j = 0 , 1 , ..., n but P n is a polynomial of degree n , it cannot have n + 1 zeros unless P n ( x ) 0, which implies a 0 = a 1 = · · · = a n = 0. That is, the homogenous problem associated with (2) has only the trivial solution.