hw3 - norm dened by || f || H = p ( f,f ) H . (It may be...

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Math 128a, fall 2011, Chorin, theory homework 3, due in the week of Sept. 19. 1. Verify that the inner product ( f,g ) W = R b a f ( x ) g ( x ) W ( x ) dx (where W ( x ) is a continuous function that is positive on [ a,b ]) satisfies all the axioms for an inner product, and furthermore that || f || W = p ( f,f ) W satisfies all the axioms for a norm. 2. Find polynomials H 0 ,H 1 ,H 2 , respectively of degrees 0 , 1 , 2, orthonormal with respect to the inner product ( f,g ) H = R infty f ( x ) g ( x ) e - x 2 / 2 2 π . Use these polynomials (they are called the “Hermite” polynomials) to find the polynomial of degree 2 that best approximates f ( x ) = x 3 in the
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Unformatted text preview: norm dened by || f || H = p ( f,f ) H . (It may be helpful to know that R - e-x 2 / 2 dx = .) 3. Find coecients a-2 ,a-1 ,a ,a 1 such that a-2 f ( x-2 h ) + a-1 f ( x-h ) + a f ( x ) + a 1 f ( x + h ) approximates f ( x ) with an error 0( h 3 ). expressions you need well dened). 4. Let f be 4 times dierentiable with a bounded 4-th derivative. Show that h-2 ( f ( x + h ) + f ( x-h )-2 f ( x )) = f 00 ( x ) + O ( h 2 ) ( f 00 is a second derivative). 1...
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This note was uploaded on 01/21/2012 for the course MATH 128A taught by Professor Rieffel during the Fall '08 term at University of California, Berkeley.

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