convolutionlet fg (rd be real or complex valued functions their convolutionf gis defined as )(xf (yg( ydyrd and is also in (rd moreoverthe fourier transform satisfies gf [ dualitylet and both be in (rd then [ ]](tf (- product formulalet fg (rd and denote by fe their respective fourier transforms then gf (rd )and (zg(zdz ( ) (zdz rd rd there are many additional properties which hold if (rd (rd in particularif fg (rd (rd )then , (rd and fe gi fgia result often known as parseval' formula putting gives the result often referred to as plancherel' theorem the fourier transform can be extended in several waysin the first instance to functions in (rd by continuity substantial extension of the theory is realized by replacing integration with respect to the lebesgue measure ( rd dxwith integration with respect to (finite borelmeasure ( rd (dx)moreoverthere is close connection between the fourier transform and characteristic functions arising in probability theory indeedif is random vector with pdf then its characteristic function ps satisfies ps( : ei ](- /( ) discrete fourier transform herewe introduce the (univariatediscrete fourier transformwhich can be viewed as special case of the fourier transform introduced in definition where integration is with respect to the counting measureand ( for ( definition discrete fourier transform discrete fourier transform the discrete fourier transform (dftof vector [ xn- ]cn is the vector [ xn- ]whose elements are given by xt - = st where exp(- /nin other wordse is obtained from via the linear transformation fx(