hw5 - y (0) = 1, as well as the corresponding Euler...

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Math 128a, fall 2011, Chorin, theory homework 5, due the week of Oct. 3. 1. Suppose P 0 ,P 1 ,P 2 ,... are polynomials orthonormal with respect to the inner product ( f,g ) = R b a f ( x ) g ( x ) W ( x ) dx , where W ( x ) > 0 is a weight function and P n is of degree n . Is it still true that P n has n distinct roots in ( a,b )? (explain your answer, ”yes” or ”no” do not suffice). 2. Show that the the Chebysheff polynomials T n = cos ( ncos - 1 ( x )) are or- thogonal on [ - 1 , +1] with respect to the inner product defined by ( f,g ) = R b a f ( x ) g ( x ) W ( x ) dx where W ( x ) = (1 - x 2 ) - 1 / 2 . 3. Which of the following functions are Lipshitz continuous on the intervals indicated: (i) f ( x ) = x 2 on [0 , 1]; (ii) f ( x ) = x 2 on [0 , ); (iii) f ( x ) = (1 - x 2 ) - 1 / 2 on [0 , 1]; (iv) f ( x ) = (1 - x 2 ) - 1 / 2 on [ - 1 / 2 , +1 / 2] (v) f ( x ) = 2 x (1 + 1 logx ) on [1 , ). 4. Sketch the slopes defined in the x,y plane by the differential equation y 0 = y ; sketch on the same diagram the solution that satisfies the condition
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Unformatted text preview: y (0) = 1, as well as the corresponding Euler polygons for h = 0 . 2 and h = . 1. 5. (A rather hard problem) Consider the dierential equation y =-y,y (0) = 1. Approximate it by the dierence equations u = 1 ,u 1 = u-hu , and u i +1-u i-1 =-2 hu i for i > 1 ( h is the mesh size). Look for exact (analytic, non-numerical) solutions of the dierence equation of the form u i = i , and nd values of by substitution into the dierence equation. You should obtain two solutions for , say 1 , 2 . Show that for one of them, (call it 1 ), i 1 tends to e-x as h tends to zero, as it should; what does i 2 do? ( 2 is the other value of ). 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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