definition (multivariatefourier transform the fourier transform of (realor complex-valuedfunction (rd is the function defined as ( : - (xdx rd fourier transform rd the fourier transform is continuousuniformly bounded (since (rd imr plies that ( ) rd ( )dx )and satisfies limktke ( ( result known as the riemann-lebesgue lemmahowevere does not necessarily have finite integral simple example in is the fourier transform of ( {- / / then (tsin(pt)/(ptsinc(pt)which is not absolutely integrable definition (multivariateinverse fourier transform the inverse fourier transform - of (realor complex-valuedfunction ( is the function fv defined as fv( :ei (tdtx rd rd as one would hopeit holds that if and are both in (rd )then - [ ]almost everywhere the fourier transform enjoys many interesting and useful propertiessome of which we list below linearityfor fg (rd and constants ab rf [ bga [ space shifting and scalinglet rdxd be an invertible matrix and rd constant vector let (rd and define ( : (ax bthen - ) [ ](tei ( ( - )/det( )|where -:( )- ( - ) frequency shifting and scalinglet rdxd be an invertible matrix and rd constant vector let (rd and define - ( : - ( - )/det( )then [ ](te (at differentiationlet (rd (rd and let fk : /xk be the partial derivative of with respect to xk if fk (rd for dthen fk ]( ( tk (tinverse fourier transform