Exploring Fourier Transforms and Integrability for Different Functions

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SUMMARY

The discussion centers on proving that the function f(x) = (1 + |x|)^{-a} serves as the Fourier transform of an integrable function on R when a > 1. It is established that for 0 < a <= 1, the integral does not converge, indicating that the Fourier transform fails in this range. Additionally, the function f(x) = 1/(log(|x|^2 + 2)) is also examined, with the conclusion that its integrability requires further analysis. The necessity for the denominator to have a power greater than 1 is emphasized to ensure convergence during the inverse transform.

PREREQUISITES
  • Understanding of Fourier transforms and their properties
  • Knowledge of integrable functions and convergence criteria
  • Familiarity with mathematical analysis concepts, particularly limits and asymptotic behavior
  • Basic proficiency in handling logarithmic functions and their integrability
NEXT STEPS
  • Study the conditions for integrability of functions in the context of Fourier transforms
  • Explore the implications of the decay rate of functions on their Fourier transforms
  • Investigate the properties of logarithmic functions and their behavior at infinity
  • Learn about the implications of the Riemann-Lebesgue lemma on Fourier transforms
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Mathematicians, physics students, and anyone involved in signal processing or harmonic analysis who seeks to deepen their understanding of Fourier transforms and integrability conditions.

shoplifter
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prove that f(x) = (1 + |x|)^{-a} is the Fourier transform of some integrable function on R, when a > 1. what happens when 0 < a <= 1? how about the function f(x) = 1/(log(|x|^2 + 2))?
 
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Is this homework? If so, what have you done so far?
 
yes, and i realize that we want the integral to converge when we take the inverse transform. so in order to do that, I'm guessing the denominator has to have a power > 1, which is why we have that condition on a. so it will fail the second time, i guess. but i can't formalize my argument (and I'm clueless about the third one).

sry :(
 
any help please? :(
 

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