Practice Conditions For Fourier Integrability (9.5)

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6 sample questions

Question 1.
What does it mean for a function to be piecewise continuous?
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Think about how many breaks a function can have.
Question 2.
Give an example of an absolutely integrable function.
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Consider functions that converge towards zero.
3Question 3.
What is a necessary condition for Fourier integrability?
- Piecewise Differentiability
- Absolute Integrability
- Periodic Behavior
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Think about what distinguishes integrable functions.
4Question 4.
True or False? A function can have infinite discontinuities and still be Fourier integrable.
- True
- False
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Recall the conditions discussed.
Question 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.
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Evaluate the behavior around 0 and its integral over the relevant range.
Question 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.
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Consider the density of rational numbers in any interval.

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