9.5. Conditions for Fourier Integrability

Try these first

1. 1.

   What does it mean for a function to be piecewise continuous?

   Hint

   Think about how many breaks a function can have.

2. 2.

   Give an example of an absolutely integrable function.

   Hint

   Consider functions that converge towards zero.

3. 3.

   What is a necessary condition for Fourier integrability?

   • Piecewise Differentiability
   • Absolute Integrability
   • Periodic Behavior

   Hint

   Think about what distinguishes integrable functions.

4. 4.

   True or False? A function can have infinite discontinuities and still be Fourier integrable.

   • True
   • False

   Hint

   Recall the conditions discussed.

5. 5.

   Analyze the function f(x) = |sin(x)/x| for x != 0 and f(x) = 0 at x = 0. Is it Fourier integrable? Discuss the conditions.

   Hint

   Evaluate the behavior around 0 and its integral over the relevant range.

6. 6.

   Demonstrate that the function f(x) defined as 1 for rational x and 0 for irrational x fails to meet the criteria for Fourier integrability.

   Hint

   Consider the density of rational numbers in any interval.