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220-hw5s

# 220-hw5s - HOMEWORK 5 MATH 220 Problem 1(i(Ffa = = Rn Rn...

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HOMEWORK 5 MATH 220 Problem 1 (i) ( F f a )( ξ ) = Z R n e - ix · ξ f a ( x ) dx = Z R n e - ix · ξ f ( x - a ) dx = Z R n e - i ( y + a ) · ξ f ( y ) dy = e - ia · ξ Z R n e - ix · ξ f ( x ) dx = e - ia · ξ ( F f )( ξ ) . (ii) ( F g a )( ξ ) = Z R n e - ix · ξ g a ( x ) dx = Z R n e - ix · ξ e ix · a f ( x ) dx = Z R n e - ix · ( ξ - a ) f ( y ) dy = ( F f )( ξ - a ) . (iii) ( F - 1 f a )( x ) = (2 π ) - n Z R n e ix · ξ f a ( ξ ) = (2 π ) - n Z R n e ix · ξ f ( ξ - a ) = (2 π ) - n Z R n e ix · ( ξ + a ) f ( y ) dy = e ia · x (2 π ) - n Z R n e ix · ξ f ( ξ ) = e ia · x ( F - 1 f )( x ) . (iv) ( F - 1 g a )( x ) = (2 π ) - n Z R n e ix · ξ g a ( ξ ) = (2 π ) - n Z R n e ix · ξ e ia · ξ f ( x ) = (2 π ) - n Z R n e i ( x + a ) · ξ f ( y ) dy = ( F - 1 f )( x + a ) . Problem 2 Let x = ( x 1 ,...,x j ,...,x n ) and h j = (0 ,...,h,. .., 0) (0 everywhere except the j th coordinate.) ( F ( j f ))( ξ ) = Z R n e - ix · ξ ( j f )( x ) dx

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220-hw5s - HOMEWORK 5 MATH 220 Problem 1(i(Ffa = = Rn Rn...

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