hw9sol - Honors Introduction to Analysis I Homework IX...

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Unformatted text preview: Honors Introduction to Analysis I Homework IX Solution April 21, 2009 Problem 1 Suppose f is C 1 on an interval and that f satisfies the Holder condition of order , | f ( x )- f ( y ) | M | x- y | for all x,y in the interval, where (0 , 1] is fixed. Show that | 2 h f ( x ) | c | h | 1+ . How does the constant c relate to M ? Solution. We apply the mean value theorem to g ( t ) = f ( t + h )- f ( t ) on the interval [ x,x + h ]. Then 2 h f ( x ) = f ( x + 2 h )- 2 f ( x + h ) + f ( x ) = f ( x + 2 h )- f ( x + h )- ( f ( x + h )- f ( x )) = g ( x + h )- g ( x ) and g ( t ) = f ( t + h )- f ( t ). By the MVT we get that g ( x + h )- g ( x ) = hg ( x ) for some x ( x,x + h ), which is equivalent to saying 2 h f ( x ) = h ( f ( x + h )- f ( x )) so | 2 h f ( x ) | = | h || f ( x + h )- f ( x ) | | h | M | x + h- x | = M | h | 1+ c | h | 1+ for c M . Hence | 2 h f ( x ) | c | h | 1+ for c M (the minimal value for c , for which this is true, is M ) Problem 2 Suppose that f has a zero of order j at x and that g has a zero of order k at x . What can you say about the order of zero of the function f + g , f g and f/g at x ? Solution. By definition f ( x ) = a j ( x- x ) j + o ( | x- x | j ) and g ( x ) = a k ( x- x ) j + o ( | x- x | k ), or f ( n ) ( x ) = 0 for n = 0 ,...,j- 1 and f ( j ) ( x ) 6 = 0 and g ( n ) ( x ) = 0 for n = 0 ,...,k- 1 and g ( k ) ( x ) 6 = 0. Let m = min { j,k } , then ( f + g ) ( n ) ( x ) = 0 for n = 0 ,...,m- 1 and ( f + g ) ( m ) ( x ) 6 = 0. Or ( f + g )( x ) = a m ( x- x ) m + o ( | x- x | m ), so f + g has a zero of order min { j,k } at x . For any n , ( fg ) ( n ) ( x ) = n i =0 a i f ( i ) ( x ) g ( n- i ) ( x ) for a i = n i . From this we notice that ( fg ) ( n ) ( x ) = 0 for n = 0 ,...,k + j- 1 and ( fg ) ( k + j ) ( x ) = k + j j f ( j ) ( x ) g ( k ) ( x ) 6 = 0, so fg has a zero of order j + k . Equivalently one can see that ( fg )( x ) = a k a j ( x- x ) j + k + o ( | x- x | k + j ). Lets first analyze the case when j < k ....
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This note was uploaded on 09/22/2010 for the course MATH 413 at Cornell University (Engineering School).

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hw9sol - Honors Introduction to Analysis I Homework IX...

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