# hw7 - Homework 7 Foundations of Computational Math 2 Spring...

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Unformatted text preview: Homework 7 Foundations of Computational Math 2 Spring 2011 Solutions will be posted 3/7/11 Problem 7.1 For this problem, consider the space L 2 [ − 1 , 1] with inner product and norm ( f, g ) = integraldisplay 1- 1 f ( x ) g ( x ) dx and bardbl f bardbl 2 = ( f, f ) Let P i ( x ), for i = 0 , 1 , . . . be the Legendre polynomials of degree i and let n + 1 − st have the form P n +1 ( x ) = ρ n ( x − x )( x − x 1 ) ··· ( x − x n ) i.e., x i for 0 ≤ i ≤ n are the roots of P n +1 ( x ). Let the Lagrange interpolation functions that use the x i be ℓ i ( x ) for 0 ≤ i ≤ n . So, for example, L n ( x ) = ℓ ( x ) f ( x ) + ··· + ℓ n ( x ) f ( x n ) is the Lagrange form of the interpolation polynomial of f ( x ) defined by the roots. Let P n be the space of polynomials of degree less than or equal to n . We can write the least squares approximation of f ( x ) in terms of the P i ( x ) using the generalized Fourier series as f n ( x ) = α P ( x ) + α 1 P 1 ( x ) +...
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hw7 - Homework 7 Foundations of Computational Math 2 Spring...

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