University Mathematics Handbook (2015)

XIII. Fourier Series and Integral Transforms

Chapter 2. Fourier Integral and Fourier Transform

Let be set of all piecewise continuous functions defined on all with values in , and absolutely integrable, that is, .

2.1  Fourier Integral

a.  If and , then, for all points of continuity of , then holds:

(*)

where

Integral (*) is called the Fourier integral of .

b.  Fourier integral (*) can be written this way:

c.  The complex form of Fourier integral is

2.2  Fourier Transform (Definitions)

a.  

is called the Fourier transform of .

b.  If , then is defined for every , continuous on and .

c.  

is called an inverse Fourier transform.

d.  Theorem: If , then for every , then has one-sided derivatives and holds

2.3  Properties and Formulas of Fourier Transform

a.  Linearity:

b.  If , then

c.  Displacement formula: if then:

1.  

2.  

3.  Modulation formulas:

d.  Derivative formula: If function is continuous, and , then

e.  If and , then Fourier transform of is continuously differentiable and there holds

f.  Plancherel formula: If and , then

and there holds

g.  Generalized Plancherel Formula:

If , and , then

h.  A convolution of two functions is

i.  

2.4  Fourier Transforms Table