10.1.1. Definition

Try these first

1. 1.

   What is the definition of the Fourier Cosine Transform?

   Hint

   Think about how this transform relates to frequency.

2. 2.

   What is the inverse Fourier Cosine Transform used for?

   Hint

   What operation helps us revert to the original function?

3. 3.

   What does the Fourier Cosine Transform decompose?

   • Sine functions
   • Cosine functions
   • Exponential functions

   Hint

   Think about the type of basis functions used.

4. 4.

   True or False: The inverse Fourier Cosine Transform is used to return to the spatial domain.

   • True
   • False

   Hint

   Consider what transforms lead to the original function.

5. 5.

   Calculate the Fourier Cosine Transform of the function $f(x) = x^2$ over [0, ∞).

   Hint

   Consider integration by parts for this problem.

6. 6.

   Examine the effects of changing parameters in the FCT with different exponential functions. What does it tell you about behavior in the frequency domain?

   Hint

   Analyze how the decay rate affects the amplitude in the frequency domain.