11.2. Conditions for Existence (Dirichlet’s Conditions)

Try these first

1. 1.

   What does it mean for a function to be absolutely integrable?

   Hint

   Think about the convergence of the integral.

2. 2.

   What is one example of a condition for function existence?

   Hint

   Consider the behavior of discontinuities.

3. 3.

   Which of the following is a condition for the existence of a Fourier Transform?

   • Infinite discontinuities
   • Absolutely integrable
   • Non-integrable function

   Hint

   Recall the integral criteria.

4. 4.

   True or False: A function can have an infinite number of maxima and still have a Fourier Transform.

   • True
   • False

   Hint

   Think about the conditions we discussed.

5. 5.

   Consider the function f(t) = 1/t for t > 1 and 0 for t <= 1. Analyze whether this function meets Dirichlet's Conditions.

   Hint

   Evaluate both endpoints of the function interval.

6. 6.

   Given f(t) = cos(t^2), describe whether it meets Dirichlet’s Conditions within the interval [-T, T].

   Hint

   Assess the frequency of oscillation and consider integration over the range.