Uncovering the Mystery of Using Cosine Transform in Fourier Analysis

In summary, the book presents a problem with the sine Fourier transform of function u, but cannot directly take the inverse transform due to a singularity at omega = 0. To solve this, the book takes the derivative of the transform and performs an inverse cosine transform on the exponential term using convolution. This is because when using convolution on sine/cosine, both must be used.
  • #1
member 428835
hi pf!

My book presents a problem and has it boiled down to $$S(u) = -S(f(x)) \exp(- \omega y) / \omega$$ where ##S(u)## is the sine Fourier transform of the function ##u##. However, we cannot directly take the transform back since the singularity at ##\omega = 0##. Thus the book then takes $$\frac{\partial}{\partial y} S(u) = S(f(x)) \exp(- \omega y)$$ and now performs an inverse COSINE transform on the exponential (also they use convolution). My question is, why are they using a cosine transform instead of a sine transform?

Thanks so much for your help!
 
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  • #2
nevermind, i got it! when using convolution on sine/cosine we use one of each!
 

What is the Fourier transform problem?

The Fourier transform problem is a mathematical concept that involves breaking down a signal or function into its individual frequencies. It is often used in signal processing, image processing, and other branches of science and engineering.

What is the purpose of the Fourier transform?

The purpose of the Fourier transform is to analyze and understand the frequency components of a signal or function. It allows us to see which frequencies are present and how much of each frequency is present in the original signal.

What is the difference between the Fourier transform and the inverse Fourier transform?

The Fourier transform converts a signal from the time domain to the frequency domain, while the inverse Fourier transform converts the signal back from the frequency domain to the time domain. Essentially, they are inverse operations of each other.

What are some applications of the Fourier transform?

The Fourier transform has many applications in various fields, including signal processing, image processing, data compression, and spectral analysis. It is also used in solving differential equations and in the study of wave phenomena.

Can the Fourier transform be applied to any signal or function?

Yes, the Fourier transform can be applied to any signal or function, as long as it is well-defined. However, the results may not always be meaningful or useful, depending on the nature of the signal or function.

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