FOURIER TRANSFORM

“The thorough study of nature is the most fertile ground for mathematical discoveries.”

Joseph Fourier

Fourier’s treatise provided the modern mathematical theory of heat conduction, Fourier series, and Fourier integrals with applications. In his treatise, Fourier stated a remarkable result that is universally known as the Fourier Integral Theorem. He gave a series of examples before stating that an arbitrary function defined on a finite interval can be expanded in terms of trigonometric series which is now universally known as the Fourier series.

Basic Concepts and Definitions

The integral transform of a function f(x) defined in a≤x≤b is denoted by I {f(x)}=F(k), and defined by

where K(x, k), given function of two variables x and k, is called the kernel of the transform. The operator I is usually called an integral transform operator or simply an integral transformation. The transform function F(k) is often referred to as the image of the given object function f(x), and k is called the transform variable. Similarly, the integral transform of a function of several variables is defined by

where x=(x1, x2, . . . , xn), κ=(k1, k2, . . . , kn), and S ⊂Rn. A mathematical theory of transformations of this type can be developed by using the properties of Banach spaces. From a mathematical point of view, such a program would be of great interest, but it may not be useful for practical applications. Our goal here is to study integral transforms as operational methods with special emphasis to applications. The idea of the integral transform operator is somewhat similar to that of the well known linear differential operator, D ≡ d/dx , which acts on a function f(x) to produce another function f'(x), that is,

Usually, f'(x) is called the derivative or the image of f(x) under the linear transformation D.

The Fourier Integral Formulas

A function f(x) is said to satisfy Dirichlet’s conditions in the interval −a< x<a, if
(i) f(x) has only a finite number of finite discontinuities in −a<x<a and has no infinite discontinuities.
(ii) f(x) has only a finite number of maxima and minima in −a<x<a.

DEFINITION. The Fourier transform of f(x) is denoted by F{f(x)} = F(k), k ∈R, and defined by the integral

where F is called the Fourier transform operator or the Fourier transformation and the factor 1/√2π is obtained by splitting the factor 1/2π. This is often called the complex Fourier transform. A sufficient condition for f(x) to have a Fourier transform is that f(x) is absolutely integrable on (−∞,∞). The convergence of the integral follows at once from the fact that f(x) is absolutely integrable. In fact, the integral converges uniformly with respect to k.

The inverse Fourier transform, denoted by F⁻¹{F(k)}=f(x), is defined by

where F⁻¹ is called the inverse Fourier transform operator.

Clearly, both F and F−1 are linear integral operators. In applied mathematics, x usually represents a space variable and k(= 2πλ ) is a wavenumber variable where λ is the wavelength. However, in electrical engineering, x is replaced by the time variable t and k is replaced by the frequency variable ω(= 2πν) where ν is the frequency in cycles per second. The function F(ω)=F{f(t)} is called the spectrum of the time signal function f(t). In electrical engineering literature, the Fourier transform pairs are defined slightly differently by

where ω =2πν is called the angular frequency. The Fourier integral formula implies that any function of time f(t) that has a Fourier transform can be equally specified by its spectrum.

Basic Properties of Fourier Transforms

If F{f(x)}=F(k), then

PROOF OF THESEWILL BE SHORTLY UPLOADED.