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# hw11 - < c ≤ 2 is sufﬁciently small then their is a...

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Math508, Fall 2010 Jerry L. Kazdan Problem Set 11 D UE : Never Note: We say a function is smooth if its derivatives of ball orders exist and are continuous. 1. Partition [ a , b ] R into sub-intervals a < x 1 < x 2 < ··· < x n = b . A function h ( x ) that is constant on each sub-interval is called a step function . Show that if f C ([ a , b ]) , then it can be approximated arbitrarily closely (in the uniform norm) by a step function. 2. Let f C ([ - 1 , 1 ]) be an even function (so f ( - x ) = f ( x ) ). Show it can be approximated arbitrarily closely (in the uniform norm) by an even polynomial. 3. Let f C 1 ([ 0 , 2 ]) . Given any ε > 0 show there is a polynomial p ( x ) such that max x [ 0 , 2 ] | f ( x ) - p ( x ) | + max x [ 0 , 2 ] | f ( x ) - p ( x ) | < ε That is, f - p C 1 ([ 0 , 2 ]) < ε . 4. a) Give an example of a continuous function f : ( 0 , 1 ] ( 0 , 1 ] that has no fixed points. b) Let A ( 1 , 3 ) and f ( x ) : = ( x / 2 ) + ( A / 2 x ) . Show that f satisfies the hypotheses of the Contracting Mapping Principle on the domain [ 1 , ) . What is the fixed point? 5. Let h ( x , y ) and f ( x ) be continuous for 0 x 2, 0 y 2. a) Show that if 0
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Unformatted text preview: < c ≤ 2 is sufﬁciently small, then their is a continuous function u ( x ) that satisﬁes u ( x ) = f ( x )+ Z c h ( x , y ) u ( y ) dy . (*) b) In the special case where h ( x , y ) ≡ 1 and f ( x ) ≡ 1, solve equation (*) explicitly. [This is easy. Let α = R 1 u ( y ) dy and then use (*) to solve for α ]. From this, show that indeed for some value of c a solution may not exist. 6. Let f ( x ) and h ( x , y ) be as in the previous problem. Show that if λ > 0 is sufﬁciently small, the equation u ( x ) = f ( x )+ λ Z 2 h ( x , y ) u ( y ) dy . (1) has a unique continuous solution u ( x ) . 7. Let f be an even continuous function on [-1 , 1 ] with R 1-1 f ( x ) x n dx = 0 for all even n ≥ 0. Show that f ≡ 0. [Last revised: December 8, 2010] 1...
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