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Unformatted text preview: MAT 330: Fourier Series and Partial Differential Equations Professor Sergiu Klainerman Problem Set 7 Rik Sengupta email@example.com April 23, 2010 Hadamards Parties Finies or Pseudo-Functions, 1932 1. Let <- 1 be a given real number and / Z . Prove that, for all the functions C ( R ), we can write ( is a small positive number) Z x ( x ) dx = P ( ) + R ( ) where P ( ) is a finite linear combination of negative powers of and R ( ) has a limit when goes to 0. Solution. We have to integrate by parts n times, where n N is the smallest positive integer such that + n >- 1 [recall that / Z ]. Then, after integrating, we have Z x ( x ) dx = (- 1) n Z x + n ( + 1) ... ( + n ) ( n ) ( x ) dx + n- 1 X j =0 A j ( j ) (0) +1+ j + o (1) , where for the last term, we have used a Taylor expansion to express derivatives of at in terms of the derivatives of at 0. Then, in order to complete the proof, take P ( ) = n- 1 X j =0 A j ( j ) (0) +1+ j + o (1) R ( ) = (- 1) n Z x + n ( + 1) ... ( + n ) ( n ) ( x ) dx. 2. Prove that the formula h pf ( x + ) , i = lim R ( ) defines a distribution on R . Solution. Take any compact set K R , and C ( R ), with supp( ) K . 1 We may assume without loss of generality that K = [0 ,M ] for some M R . Then, it follows that h pf ( x + ) , i = Z x + n ( + 1) ... ( + n ) ( n ) ( x ) dx C sup ( n ) ( x ) Z M x + n dx = C sup ( n ) ( x ) , where the supremum runs over the compact set K . Here we have chosen bounds C and C based on properties of compact sets and continuous functions. This calculation shows that pf ( x + ) defines a distribution on R ....
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