1. Fourier transforms represent a function as a sum of sinusoidal functions using integral transforms. The Fourier transform of a function f(x) is defined as an integral transform using a kernel function, with examples including the Laplace, Fourier, Hankel, and Mellin transforms.
2. The Fourier integral theorem states that if a function f(x) is piecewise continuous and differentiable, its Fourier transform represents the function as an integral using sinusoidal functions.
3. The Fourier transform and its inverse are defined by integrals using the function and a complex exponential kernel. Properties of Fourier transforms include linearity, shifting, scaling, and relationships between a function and its derivative or integral transforms.
3. FOURIER INTEGRAL
THEOREM
If ๐(๐ฅ) is piece-wise continuously, differentiable and
absolutely integrable in (โโ, โ)
Then ๐ ๐ฅ =
1
2๐ โโ
โ
โโ
โ
๐ ๐ก ๐ ๐๐ ๐ฅโ๐ก ๐๐ก๐๐
Or ๐ ๐ฅ =
1
๐ 0
โ
โโ
โ
๐ ๐ก cos ๐ ๐ก โ ๐ฅ ๐๐ก๐๐
This is called Fourier integral theorem/Formula.
4. FOURIER TRANSFORM
If ๐(๐ฅ) is defined in(โโ, โ), then the
Fourier Transform of ๐(๐ฅ) is given by
๐น ๐ ๐ฅ = ๐น ๐ =
1
2๐ โโ
โ
๐ ๐ฅ ๐ ๐๐ ๐ฅ
๐๐ฅ
6. Fourier Sine Transform
The Fourier Sine Transform is given by
๐น๐ ๐ ๐ฅ = ๐น๐ ๐ =
2
๐ 0
โ
๐ ๐ฅ ๐ ๐๐๐ ๐ฅ ๐๐ฅ
Inverse Fourier Sine Transform
The Inverse Fourier Sine Transform of ๐น๐ ๐ is given by
๐ ๐ฅ =
2
๐ 0
โ
๐น๐ (๐ )๐ ๐๐๐ ๐ฅ ๐๐
Fourier Cosine Transform
The Fourier Cosine Transform is given by
๐น๐ ๐ ๐ฅ = ๐น๐ ๐ =
2
๐ 0
โ
๐ ๐ฅ ๐๐๐ ๐ ๐ฅ ๐๐ฅ
Inverse Fourier Cosine Transform
The Inverse Fourier Cosine Transform of ๐น๐ ๐ is given
by
๐ ๐ฅ =
2
๐ 0
โ
๐น๐(๐ )๐๐๐ ๐ ๐ฅ ๐๐
8. CONVOLUTION OF TWO
FUNCTIONS
The convolution of two functions ๐(๐ฅ)
and ๐(๐ฅ)
is defined by
๐ โ ๐ =
1
2๐ โโ
โ
๐ ๐ก ๐ ๐ก โ ๐ฅ ๐๐ก
9. Convolution Theorem
๐น(๐ โ ๐) = ๐น(๐ ). ๐บ(๐ ) where ๐ and ๐ are two
functions and ๐น(๐ ) =
1
2๐ โโ
โ
๐ ๐ฅ ๐ ๐๐ ๐ฅ
๐๐ฅ
& ๐บ(๐ ) =
1
2๐ โโ
โ
๐ ๐ฅ ๐ ๐๐ ๐ฅ
๐๐ฅ
Parsevalโs Identity
A function f(x) and its transform F(s) satisfy the
identity
โโ
โ
๐ ๐ฅ 2
๐๐ฅ = โโ
โ
๐น ๐ 2
๐๐