9.15. Exercises

Try these first

1. 1.

   What is the Fourier integral formula for f(x) = xe^{-x}, x > 0?

   Hint

   Consider applying integration by parts.

2. 2.

   What bounds do we use when evaluating the Fourier sine integral of a step function?

   Hint

   Focus on where f(x) is defined.

3. 3.

   What is the main purpose of the Fourier cosine integral?

   • To analyze periodic functions
   • To represent non-periodic functions
   • To calculate Fourier coefficients

   Hint

   Recall the main objective of Fourier transforms.

4. 4.

   True or False: The Fourier sine integral is defined only for odd functions.

   • True
   • False

   Hint

   Think about the properties of sine functions.

5. 5.

   Using the Fourier integral representation, prove the Parseval's theorem for a function defined over an infinite domain.

   Hint

   Start by expressing both sides using the integral definitions.

6. 6.

   Consider the function f(x) = e^{-a|x|}. Derive its full Fourier transform and discuss the significance of the results obtained in signal processing.

   Hint

   Use properties of even functions to aid in the derivation.