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 . De±ne the partial Fourier transform of f by ( F z f )( y, ζ ) = i R k eiz · ζ f ( y, z ) dz, y ∈ R m , ζ ∈ R k . Show that (i) ( F z D z j f )( y, ζ ) = ζ j ( F z f )( y, ζ ). (ii) ( F z D y j f )( y, ζ ) = ( D y j ( F z f ))( y, ζ ). Note : Under appropriate additional assumptions, as for the full Fourier transform, the formulae F z ( z j f ) = − D ζ j F z f , F z ( y j f ) = y j F z f , and the analogous formulae for ( F1 ζ ψ )( y, z ) = (2 π )k i R k e iz · ζ ψ ( y, ζ ) dz also hold, but you do not need to prove this (but you should know these!). 1...
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 '09
 Math, Differential Equations, Applied Mathematics, Equations, Partial Differential Equations, Fourier Series, Heaviside step function, Rectangular function

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