Inverse Fourier transform as an integral
Inverse Fourier transform as an integral [ edit ] The most common statement of the Fourier inversion theorem is to state the inverse transform as an integral. For any integrable function � and all � ∈ � � set � − 1 � ( � ) := ∫ � � � 2 � � � ⋅ � � ( � ) � � . Then for all � ∈ � � we have � − 1 ( � � ) ( � ) = � ( � ) . Fourier integral theorem [ edit ] The theorem can be restated as � ( � ) = ∫ � � ∫ � � � 2 � � ( � − � ) ⋅ � � ( � ) � � � � . If f is real valued then by taking the real part of each side of the above we obtain � ( � ) = ∫ � � ∫ � � cos ( 2 � ( � − � ) ⋅ � ) � ( � ) � � � � . Inverse transform in terms of flip operator [ edit ] For any function � define the flip operator [note 1] � by � � ( � ) := � ( − � ) . Then we may instead define � − 1 � := � � � = � � � . It is immediate from the definition of the Fourier transform and the flip operator that both � � � and � � � ...