6 - Fourier Transform - Basic Properties and the Inversion Formula

6 - Fourier Transform - Basic Properties and the Inversion Formula

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Unformatted text preview: MATH 220: THE FOURIER TRANSFORM BASIC PROPERTIES AND THE INVERSION FORMULA The Fourier transform is the basic and most powerful tool in studying constant coefficient PDE on R n . It is based on the following simple observation: for R n , the functions v ( x ) = e ix = e ix 1 1 e ix n n are joint eigenfunctions of the operators x j , namely for each j , (1) x j v = i j v . It would thus be desirable to decompose an arbitrary function u as an (infinite) linear combination of the v , namely write it as (2) u ( x ) = (2 ) n integraldisplay R n u ( ) e ix d, where u ( ) is the amplitude of the harmonic e ix in u . (The factor (2 ) n is here due to a convention, it could also be moved to other places.) It turns out that this identity, (2), holds provided that we define u ( ) = integraldisplay R n e ix u ( x ) dx ; (2) is then the Fourier inversion formula . Rather than showing this at once, we start with a step-by-step approach. We first define the Fourier transform as ( F u )( ) = integraldisplay R n e ix u ( x ) dx for u C ( R n ) with | x | N | u | bounded for some N > n (i.e. | u ( x ) | M | x | N for some M in say | x | > 1); one could instead assume simply that u L 1 ( R n ) if one is familiar with the Lebesgue integral (but actually just working with the special case mentioned above gives everything we want). Note that for such functions | ( F u )( ) | integraldisplay R n | e ix || u ( x ) | dx = integraldisplay R n | u ( x ) | dx, so F u is actually a bounded continuous function. We can similarly define the inverse Fourier transform ( F 1 )( x ) = (2 ) n integraldisplay R n e ix ( ) d ; then F 1 maps u C ( R n ) with | x | N | u | bounded for some N > n (or indeed u L 1 ( R n )) to bounded continuous functions. With these definition it is of course not clear whether F 1 is indeed the inverse of F , and worse, not even clear whether F 1 F makes sense for L 1 ( R n ), since F is then a bounded continuous function, which is not sufficient to ensure that the integral defining F 1 actually converges! We thus proceed to study properties of F and F 1 . First, we note a property of F which is the main reason for its usefulness in studying PDE, and which is an immediate consequence of (1). Namely, suppose that C 1 ( R n ) and both and 1 2 all first derivatives j , j = 1 ,...,n , decay at infinity in the same sense as above (so | x | N j is bounded for some N > n ). Then integration by parts gives ( F ( x j ))( ) = integraldisplay R n e ix x j ( x ) dx = integraldisplay R n x j ( e ix ) ( x ) dx = i j integraldisplay R n e ix ( x ) dx = i j ( F )( ) ....
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This note was uploaded on 03/16/2010 for the course CME 303 taught by Professor Vasy during the Fall '10 term at Stanford.

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6 - Fourier Transform - Basic Properties and the Inversion Formula

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