Continuous Fourier Transform of Vanishing Fast Functions: Explained

In summary, the continuous Fourier transform of a continuous function that vanishes fast enough is also a continuous function. Additionally, if the function belongs to L1(R), its Fourier Transform is uniformly continuous. L1(R) includes all functions f : R → R that are absolutely integrable.
  • #1
Delta2
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Can someone tell me if the continuous Fourier transform of a continuous (and vanishing fast enough ) function is also a continuous function?
 
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  • #2
I can tell you more: in fact, if [itex]f \in L^{1}(\mathbb R)[/itex] then its Fourier Transform is uniformly continuous.
 
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  • #3
Thanks very much but can u ... remind me which functions belong to L1(R)?
 
  • #4
Delta² said:
Thanks very much but can u ... remind me which functions belong to L1(R)?

It are all the functions ##f:\mathbb{R}\rightarrow \mathbb{R}## which are absolutely integrable. That is, for which

[tex]\int_{-\infty}^{+\infty} |f(x)|dx[/tex]

is finite (and the integral makes sense).
 
  • #5
Thx again.
 

1. What is the Continuous Fourier Transform?

The Continuous Fourier Transform is a mathematical operation that decomposes a continuous signal into its constituent frequencies. It represents the signal as a combination of sine and cosine functions at different amplitudes, frequencies, and phases.

2. How is the Continuous Fourier Transform different from the Discrete Fourier Transform?

The Continuous Fourier Transform operates on continuous signals, while the Discrete Fourier Transform operates on discrete signals. The Continuous Fourier Transform uses integrals to represent the signal in the frequency domain, while the Discrete Fourier Transform uses a finite number of samples to represent the signal.

3. What is the significance of the Fourier Transform in signal processing?

The Fourier Transform is an essential tool in signal processing as it allows us to analyze signals in the frequency domain. This helps in understanding the frequency components of a signal and identifying any noise or distortions present. It also enables us to filter out unwanted frequencies and manipulate signals for various applications.

4. How is the Continuous Fourier Transform used in image processing?

In image processing, the Continuous Fourier Transform is used to convert an image from the spatial domain to the frequency domain. This allows for the detection of patterns and features in the image, such as edges and textures. The inverse Continuous Fourier Transform is then applied to reconstruct the image in the spatial domain.

5. What are some real-world applications of the Continuous Fourier Transform?

The Continuous Fourier Transform has various real-world applications, including audio and speech processing, image and video compression, radar and sonar signal analysis, and medical imaging. It is also used in fields such as physics, engineering, and economics for data analysis and modeling.

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