§4   Fourier Transform

1. Fourier integral

[ Fourier integral ]   A function that is absolutely integrable over any finite interval [- l , l ] , its Fourier series can be found ( §1 , 2 of this chapter )

(1)

Assuming that the function is absolutely integrable over the infinite interval (- ) , in equation (1) , let l , obtain the Fourier integral of f ( x )

[ Several forms of Fourier integral ]

Let the Fourier integral satisfy the convergence condition, then

1 o =

2 o = (     outer integral is understood as integral in the sense of principal value )

3 o  is an even function:

=

4 o  is an odd function:

=

[ Convergence discriminant method of Fourier integral ] Assuming that the   function is absolutely integrable, the imaginary value of the integral (1) is S 0 . Suppose that the point x 0 is continuous, or x 0 is its first-type discontinuous point, and the continuous point is At point x 0 S 0 = , while at the first type of discontinuity point x 0 ,

S 0 =

1 o  Dini discriminant decree , if for a , the integral

converges, then the Fourier integral converges at the point x 0 and is equal to S 0 .

2 o  Dirichlet - Rawdang discriminant If there is a bounded variation on an interval [ x 0 - h , x 0 + h ] with x 0 as the midpoint, then its Fourier integral is at the point x 0 converges and is equal to S 0 .

3 o  If the function has bounded variation on, while

Then the Fourier integral converges at any point x 0 and is equal to S 0 .

2. Fourier transform

The Fourier transform of [ Fourier transform and its inversion formula ]   is

The inversion formula of the Fourier transform is

The Fourier transform and inversion formula of [ Conditions for the existence of Fourier transform ] are meaningful   ( only at the discontinuity point x 0 of ) under the following two conditions, and the left end of the inversion formula should be equal to ):

1 o  exists;

2 o  satisfies the Dirichlet condition: there are only a finite number of extreme points and only a finite number of discontinuities of the first kind .

[ Properties of Fourier Transform ]   Let the Fourier transform of g ( x ) be F ( ) and G ( ) respectively, then

1 o  The Fourier transform of a linear a + b g ( x ) is a F ( ) + b G ( ) ( a , b are constants )

2 o  The Fourier transform of convolution ( or convolution ) f ( x )*g( x )=     is

F ( ) G ( )

3 o  Parsepha equation

4 o  The Fourier transform of the flip f ( -x ) is F (- ).

The   Fourier transform of the 5o conjugate is .

The Fourier transform of 6o time - shift ( delay )    f ( x - x 0 ) is .

7o frequency  shift ( frequency modulation )    is the Fourier transform of ( is a constant ).

[ Fourier transform table ]

,

 ( g is Euler's constant )

3. Fourier cosine transform

[ Fourier cosine transform and its inversion formula ]   The Fourier cosine transform of f ( x ) is

The inversion formula of the Fourier cosine transform is:

[ Existence condition of Fourier cosine transform ]   is the same as the Fourier integral convergence condition .

[ Properties of Fourier Cosine Transform ]

1 o  If it is the Fourier cosine transform of f ( x ) , then it is the Fourier cosine transform of .

2 o  If f ( x ) is an even function, then .

The Fourier cosine transform of 3 o ( a > 0) is .

[ Fourier cosine transform table ]

,

Four, Fourier sine transform

[ Fourier sine transform and its inversion formula ]   The Fourier sine transform of f ( x ) is

The inversion formula of the Fourier sine transform is

[ Existence condition of Fourier sine transform ]   is the same as the Fourier integral convergence condition .

[ Properties of Fourier Sine Transform ]

1 o  If it is the Fourier sine transform of f ( x ) , then it is the Fourier sine transform of .

2 o  If f ( x ) is an odd function, then .

The Fourier sine transform of 3 o ( a > 0) is .

[ Fourier sine transformation table ]

,

 ( for Euler's constant )

5. Finite Fourier Cosine Transform

[ Finite Fourier cosine transform and its inversion formula ]   Let f ( x ) satisfy the Dirichlet condition in the interval ( see this section, two ) , then the finite Fourier cosine transform of f ( x ) is

The inversion formula of the finite Fourier cosine transform is:

at each successive point of f ( x ) in the interval

At the discontinuity, the left-hand side of the equation is changed to .

[ Finite Fourier Cosine Transform Table ]

,

 1 , ( m is an integer )

6. Finite Fourier Sine Transform

[ Finite Fourier sine transform and its inversion formula ]   Let f ( x ) satisfy the Dirichlet condition in the interval ( see this section, 2 ) , then the finite Fourier sine transform of f ( x ) is

The inversion formula of the finite Fourier sine transform is:

at each successive point of f ( x ) on the interval

At the discontinuity, the left-hand side of the equation is changed to .

[ Finite Fourier Sine Transform Table ]

,

 1 , ( m is an integer ) ( m is an integer )

Seven, double Fourier transform and its inversion formula

The double Fourier transform of f ( x , y ) is

The inversion formula of the double Fourier transform is: