Fourrier Tranform and schwartz space

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In summary, the conversation discusses the Fourier transform of a function f, where f belongs to the Schwartz space and k is greater than 0. It is shown that if f belongs to the Schwartz space, then f_k also belongs to the Schwartz space. Additionally, the Fourier transform of exp((-x^2)/2) is sqrt(2pi)*exp((-e^2)/2), and the first part is used to obtain the Fourier transform for exp((-ax^2)/2).
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Stephen88
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^_f=fourrier transform of f.
f_k=f<sub>k
sqrt= square root

The function f belongs to the schwartz space and k>0 f_k(x)=f(kx).
1)show that f_k also belongs to the schwartz space and ^_f(e)=(1/k)^_f(e/k)
2)the fourrier transform of exp((−x^2)/2) is sqrt(2pi)*exp((−e^2)/2) use the first part to obtain the fourrier transform for exp(−ax^2)

Attempt:
f belongs to the schwartz space then f is infinitly diff also f(kx)=kf(x) which belongs to the schwartz space.
then f_k(x)=f(kx)=kf(x) which belongs to the schwartz space.
I don't know if this is correct or how to continue...any help will be great.Thank you
 
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Should I change something ?
 

FAQ: Fourrier Tranform and schwartz space

What is the Fourier Transform?

The Fourier Transform is a mathematical tool used to decompose a signal into its constituent frequencies. It converts a signal from the time domain to the frequency domain, allowing for a better understanding of the signal's frequency components.

What is the significance of the Fourier Transform?

The Fourier Transform is significant because it allows for the analysis and manipulation of signals in the frequency domain, which can provide valuable insights and applications in various fields such as signal processing, image processing, and data compression.

What is the relationship between the Fourier Transform and Schwartz Space?

Schwartz Space, also known as the Schwartz-Bruhat Space, is a mathematical construct used to define the space of functions for which the Fourier Transform is well-defined. In other words, it is the set of functions for which the Fourier Transform exists and has specific properties.

What are some common applications of the Fourier Transform and Schwartz Space?

Some common applications of the Fourier Transform and Schwartz Space include audio and image processing, data compression, and solving differential equations.

Is the Fourier Transform limited to one-dimensional signals?

No, the Fourier Transform can be extended to higher dimensions, such as two-dimensional signals (images) and three-dimensional signals (volumes) through the use of multi-dimensional Fourier Transforms.

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