245B_W09_hwk6

245B_W09_hwk6 - 245B, Winter 2009, Assignment 6: Notes and...

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245B, Winter 2009, Assignment 6: Notes and selected model answers (Model solutions follow question numbers in bold .) Folland Chapter 4 63. We first recall that since [0 , 1] is compact, any f C [0 , 1] is bounded, and that since [0 , 1] × [0 , 1] is compact, the continuous function K : [0 , 1] × [0 , 1] C is bounded and is actually uniformly continuous [this is crucial for this question]. From the second of these observations it follows that for any ε > 0 we can choose δ > 0 so small that | K ( x,y ) - K ( x 0 ,y 0 ) | < ε whenever max {| x - x 0 | , | y - y 0 |} < δ , and so certainly when y 0 = y and | x - x 0 | < δ . Therefore given any f , using that k f k u is finite, if | x - x 0 | < δ then we also have | Tf ( x ) - Tf ( x 0 ) | = ± ± ± Z 1 0 K ( x,y ) f ( y ) d y - Z 1 0 K ( x 0 ,y ) f ( y ) d y ± ± ± Z 1 0 | K ( x,y ) - K ( x 0 ,y ) || f ( y ) | d y < ε Z 1 0 | f ( y ) | d y ε k f k . Since ε was chosen arbitrarily this shows that for any fixed f the output function Tf is continuous. Moreover, in case k f k u 1 the above inequality specializes to the assertion that | Tf ( x ) - Tf ( x 0 ) | < ε whenever | x - x 0 | < δ without any further restrictions on f . Since this
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This note was uploaded on 04/06/2010 for the course MATH various taught by Professor Tao/analysis during the Spring '10 term at UCLA.

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245B_W09_hwk6 - 245B, Winter 2009, Assignment 6: Notes and...

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