# hw5 - y(0 = 1 as well as the corresponding Euler polygons...

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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 diﬀerential equation y =-y,y (0) = 1. Approximate it by the diﬀerence 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 diﬀerence equation of the form u i = ρ i , and ﬁnd values of ρ by substitution into the diﬀerence equation. You should obtain two solutions for ρ , say ρ 1 ,ρ 2 . Show that for one of them, (call it ρ 1 ), ρ i 1 tends to e-x as h tends to zero, as it should; what does ρ i 2 do? ( ρ 2 is the other value of ρ ). 1...
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