hw2 - Math 128a, fall 2011, Chorin, theory homework 2, due...

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Unformatted text preview: Math 128a, fall 2011, Chorin, theory homework 2, due in the week of Sept. 12. 1. Show that the norm || f || satisfies all the axioms for a norm. 2. We have seen that the error in interpolation depends on the values of the polynomial Q ( x ) = ( x- x )( x- x 1 ) ... ( x- x n ), where the x i are the ( n + 1) interpolation points. Suppose there are n + 1 interpolation points in [- 1 , +1] equidistant from each other with x =- 1 and x n = 1. Conside the value of Q ( x ) at the point P half way between x and the next point x 1 . Show that this value is (2 n- 1)!! /n n +1 , (for an odd integer m , m !! is defined as m ( m- 3) ... 3 1). Deduce that if the function f ( x ) = e x is approximated by equidistant interpolation of degree n on [- 1 , 1], the max norm of the error satisfies || f- P n || e- 1 (2 n- 1)!! / ( n n +1 (( n + 1)!) , where P n is the interpolation polynomial. 3. Suppose the function f ( x ) = e x is approximated in [- 1 , 1] by the polyno- mial obtained by expanding it in Taylor series at the origin and keeping...
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