9.5 - Conditions for Fourier Integrability
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Practice Questions
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What does it mean for a function to be piecewise continuous?
💡 Hint: Think about how many breaks a function can have.
Give an example of an absolutely integrable function.
💡 Hint: Consider functions that converge towards zero.
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Interactive Quizzes
Quick quizzes to reinforce your learning
What is a necessary condition for Fourier integrability?
💡 Hint: Think about what distinguishes integrable functions.
True or False? A function can have infinite discontinuities and still be Fourier integrable.
💡 Hint: Recall the conditions discussed.
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Challenge Problems
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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.
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.
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