Fourier transform and dirichlet conditions

[Ahmad Kishki]

when a function doesn't satisfy dirichlet condition, why do we not care and go ahead finding the Fourier transform anyway? What is the use?

Eg: unit impulse, dirac delta function, etc. don't statisfy the dirichlet conditions but its like dirichlet conditions arent really conditions?

 

[jvicens]

Dear fellow scientist,

I understand your confusion about why we continue to find the Fourier transform even when a function does not satisfy the Dirichlet condition. Let me explain the reasoning behind this approach.

Firstly, the Dirichlet condition is a mathematical condition that ensures the convergence of the Fourier series of a function. This means that if a function satisfies the condition, we can accurately represent it as a sum of sinusoidal functions. However, this condition is not necessary for finding the Fourier transform of a function.

The Fourier transform is a powerful tool in signal processing and analysis. It allows us to decompose a function into its frequency components, which can be very useful in understanding the behavior of a system. Even though a function may not satisfy the Dirichlet condition, it may still have a well-defined Fourier transform.

For example, the Dirac delta function, which is not a proper function, has a very useful Fourier transform that is widely used in many applications. The Fourier transform of the unit impulse is also essential in signal processing and is used to represent the frequency response of a system.

In summary, the Dirichlet condition is not a necessary requirement for finding the Fourier transform. It is a useful condition for representing a function as a Fourier series, but it is not a limitation for finding the Fourier transform. We should not disregard the usefulness of the Fourier transform just because a function does not satisfy the Dirichlet condition.

I hope this explanation helps clear up any confusion. Keep exploring and utilizing the power of the Fourier transform in your research.