330 HW 07 - MAT 330: Fourier Series and Partial...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT 330: Fourier Series and Partial Differential Equations Professor Sergiu Klainerman Problem Set 7 Rik Sengupta rsengupt@princeton.edu 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 ....
View Full Document

Page1 / 6

330 HW 07 - MAT 330: Fourier Series and Partial...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online