Fast Fourier Transform

Abstract

If f is an integrable function of a variable $t\in (-\ ,\,\infty )$, then its Fourier integral or Fourier transform may be defined as the complex function F of a variable $\omega \in (-\infty ,\,\infty )$ where $F(\omega ):=\int _{-\infty }^{\infty }f(t)\,e^{-i\omega t}\, dt,\quad i=\sqrt{-1}$. The most important fact about definition (1) is that if it is viewed as an integral equation, its solution under very general conditions is

$$f(t)=\frac{1}{2\pi }\int _{-\infty }^{\infty }F(\omega )\,e^{i\omega t}\, d\omega (2)$$

Equation (2) is called the inverse transform of the direct transform of (1), and the two constitute the Fourier transform pair.