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**Unformatted text preview: **Problem 5. Let R n = R m R k , and write R n x = ( y, z ) R m R k . Suppose that f C 1 ( R m R k ) and | z | K f , | z | K x j f are bounded for all j = 1 , . . ., n , and K > k . Dene the partial Fourier transform of f by ( F z f )( y, ) = i R k e-iz f ( y, z ) dz, y R m , R k . Show that (i) ( F z D z j f )( y, ) = j ( F z f )( y, ). (ii) ( F z D y j f )( y, ) = ( D y j ( F z f ))( y, ). Note : Under appropriate additional assumptions, as for the full Fourier transform, the formulae F z ( z j f ) = D j F z f , F z ( y j f ) = y j F z f , and the analogous formulae for ( F-1 )( y, z ) = (2 )-k i R k e iz ( y, ) dz also hold, but you do not need to prove this (but you should know these!). 1...

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