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Unformatted text preview: 1 Lecture 11 Definition 1 The Fourier Transform of a function defined on R n is given by F ( x )( ξ ) = ˆ f ( ξ ) = 1 (2 π ) n Z R n f ( x ) e- ix · ξ dx Note: A sufficient condition for the existence of the Fourier transform is f ∈ L 1 ( R ), in which case ˆ f ∈ C ( R ), but may not belong to f ∈ L 1 ( R ). Theorem 1 If the Fourier transfrom of both f ( x ) and f ( x ) exist, then F ( f ( x ))( ξ ) = iξ ˆ f ( ξ ) More generally, if f ( x ) and all its derivatives up to order m have Fourier transforms, then F ( f ( k ) ( x ))( ξ ) = ( iξ ) k ˆ f ( ξ ) , k = 1 ,...,m If f and ˆ f ∈ L 1 ( R ), then the inverse Fourier transform is defined by F- 1 ( f )( x ) = Z R ˆ f ( ξ ) e iξ · x dξ Definition 2 A function φ ( x ) defined on R is a test function if φ is in C ∞ ( R ) and has compact support. We write φ ∈ C ∞ ( R ) . (Note: This notation varies from book to book.) Theorem 2 C ∞ ( R ) is a dense subspace of L p ( R ) , 1 ≤ p ≤ ∞ ....
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This note was uploaded on 03/14/2011 for the course MATH 676 taught by Professor Staff during the Fall '08 term at Colorado State.
- Fall '08