solhw7

solhw7 - Solutions for Homework 7 Foundations of...

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Math 2 Spring 2011 Problem 7.1 For this problem, consider the space L 2 [ 1 , 1] with inner product and norm ( f, g ) = i 1 - 1 f ( x ) g ( x ) dx and b f b 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 0 )( 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 i ( x ) for 0 i n . So, for example, L n ( x ) = 0 ( x ) f ( x 0 ) + ··· + n ( x ) f ( x n ) is the Lagrange form of the interpolation polynomial of f ( x ) de±ned 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 0 P 0 ( x ) + α 1 P 1 ( x ) + ··· + α n P n ( x ) where α i = ( f, P i ) ( P i , P i ) 7.1.a Clearly, ( i , ℓ i ) n = 0. Show that ( i , ℓ j ) = 0 when i n = j . Therefore, the functions 0 ( x ) , . . . , ℓ n ( x ) are an orthogonal basis for P n . Solution: Recall that P n +1 ( x ) P n so ( P n +1 , p ) = 0 for any p ( x ) P n . Also, we have for a constant scale factor γ i i ( x ) = γ i P n +1 ( x ) ( x x i ) We therefore have ( i , ℓ j ) = γ i γ j i 1 - 1 p P n +1 ( x ) ( x x i ) Pp P n +1 ( x ) ( x x j ) P dx = γ i γ j i 1 - 1 P n +1 ( x ) p P n +1 ( x ) ( x x i )( x x j ) P dx = γ i γ j ( P n +1 , p n - 1 ) = 0 where p n - 1 ( x ) P n since it is a polynomial of degree n 1. 1
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solhw7 - Solutions for Homework 7 Foundations of...

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