Mathquery

Definition  Of Fourier Transform :         The Fourier Transform of a function f denoted as f̂ is defined by                                  ∞       f̂ (ξ) = 1/√(2π ) ∫ f(x) e ^ -iξx dx    .....(1)                                -∞          whenever the integral on the right exists . It is obvious that the integral on the right of (1) exists if                           ∞                            ∫ |f(x)| dx    exists .                          -∞   If the fourier transform f̂ of a function f is known the function f can be obtained by the following formula , known as the inversion formula :                               ∞    f(x) = 1/√(2π )   ∫ f̂(ξ) e^iξx dξ   ..............(2)                              -∞    Sometimes it is convenient to use the operator notation F and F⁻¹ for the Fourier transform and its inverse i.e.          F(f) = f̂     and  F⁻¹(f̂) = f .    The following important properties of the Fourier transform can be easily verified