Fourier Trasform of Delta functions

In summary, the question being discussed is why the Fourier transform of a constant function is equivalent to the Dirac delta distribution. This is justified by imposing self-adjointness and using the duality pairing, as shown in the equation <F(δ), g> = < δ, g>. The link provided explains this further and provides a tutorial on the Dirac delta-function and Fourier transformation.
  • #1
muzialis
166
1
Hi All,

I am trying to understand more rigorously why the Fourier transform of a constant functions equals the Dirac delta distribution.

I found somewhere this is justified by imposing the self-adjointness, so that under a duality pairing <..,..> and indicating with F(f) the transform of a function f, it is required that

<F(δ), g> = < δ , g >

If g equals the constant unitary function my source, http://en.wikipedia.org/wiki/Dirac_delta_function#Fourier_transform, quotes,

<1, F (g)> = g (0)= < δ , g >

I understand the second equality, but not sure about the first...

Many thanks for your help
 
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  • #2

1. What is a Fourier Transform?

A Fourier Transform is a mathematical tool used to convert a function from its original domain (usually time or space) to a representation in the frequency domain. It allows us to analyze the frequency components of a function and better understand its behavior.

2. What is a Delta function?

A Delta function, also known as the Dirac delta function, is a mathematical function that is defined to be zero everywhere except at one specific point, where it is infinitely large. It is often used as a representation of an impulse or spike in a signal.

3. How is a Delta function related to the Fourier Transform?

The Fourier Transform of a Delta function is a constant function, meaning that it has the same value at all frequencies. This is because a Delta function contains all frequencies, so its Fourier Transform is a representation of all frequencies in the signal.

4. What is the significance of the Fourier Transform of a Delta function?

The Fourier Transform of a Delta function has many practical applications, such as in signal processing and image processing. It can be used to analyze the frequency components of a signal and to filter out unwanted noise or distortions.

5. Can a Delta function be transformed back to its original domain?

Yes, a Delta function can be transformed back to its original domain using the inverse Fourier Transform. This allows us to reconstruct the original signal or function from its frequency components.

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