app-c - Appendix C Integral transforms C.1 Fourier...

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Unformatted text preview: Appendix C Integral transforms C.1 Fourier transform Given a real-valued function f ( x ) on the real line, define the Fourier transform of f ( x ) to be f ( k ) = integraldisplay f ( x )exp( ikx ) dx (C.1) Then it can be shown that f ( x ) = integraldisplay dk 2 f ( k )exp( ikx ) (C.2) known as the inverse Fourier transform . Precise factors of 2 vary from source to source; what is important is that in a Fourier transform followed by an inverse Fourier transform, there should be an overall factor of 1 / (2 ). A useful identity is the following expression for the Dirac delta function: ( x ) = 1 2 integraldisplay exp( ikx ) dk (C.3) One way to derive this expression is as the inverse Fourier transform of the Fourier transform of the Dirac delta function. A useful identity is known as Parsevals theorem : integraldisplay | f ( x ) | 2 dx = 1 2 integraldisplay | f ( k ) | 2 dk This can be derived using inverse Fourier transforms: integraldisplay | f ( x ) | 2 dx = integraldisplay dx bracketleftbigg 1 2 integraldisplay f ( k ) exp(+ ikx ) dk bracketrightbiggbracketleftbigg 1 2 integraldisplay f ( k )exp( ik x ) dk bracketrightbigg = 1 (2 ) 2 integraldisplay dk integraldisplay dk f ( k ) f ( k ) integraldisplay dx exp( i ( k k ) x ) 573 APPENDIX C. INTEGRAL TRANSFORMS 574 = 1 (2 ) 2 integraldisplay dk integraldisplay dk f ( k ) f ( k )(2 ) ( k k ) = 1 2 integraldisplay | f ( k ) | 2 dk There is also a notion of convolution . Given functions f ( x ), g ( x ), define ( f g )( x ) = integraldisplay g ( y ) f ( x y ) dy It is straightforward to check that the Fourier transform of the convolution of f ( x ), g ( x ), is the ordinary product of the Fourier transforms: tildewidest ( f g )( k ) = integraldisplay ( f g )( x )exp( ikx ) dx = integraldisplay dx exp( ikx ) integraldisplay g ( y ) f ( x y ) dy = integraldisplay g ( y )exp( iky ) dy integraldisplay f ( x y )exp( ik ( x y )) dx = bracketleftbiggintegraldisplay g ( y )exp( iky ) dy bracketrightbiggbracketleftbiggintegraldisplay f ( x )exp( ikx ) dx bracketrightbigg = f ( k ) g ( k ) Similarly, if we define the convolution ( f g )( ) = 1 2 integraldisplay g ( k ) f ( k k ) dk then its inverse Fourier transform is the ordinary product of f ( x ) and g ( x ): integraldisplay ( f g )( k )exp( ikx ) dk 2 = integraldisplay dk 2 integraldisplay dk 2 g ( k ) f ( k k )exp( ikx ) = integraldisplay dk 2 g ( k )exp( ik x ) integraldisplay...
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This note was uploaded on 11/02/2011 for the course ECON 7125 taught by Professor Smith during the Spring '11 term at James Madison University.

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app-c - Appendix C Integral transforms C.1 Fourier...

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