Why Does Inverse Fourier Transform of Sinc Function Require Contour Integration?

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SUMMARY

The inverse Fourier transform of the sinc function necessitates the use of contour integration and Cauchy principal values due to the inadequacy of standard Fourier transform definitions in certain contexts. The discussion highlights that while the Fourier transform of rect(x) yields 2sinc(2πk), the inverse transform of sinc requires careful treatment to avoid incorrect results. The Cauchy Principal Value is identified as the appropriate method for evaluating the improper integral involved, emphasizing the importance of considering periodic approximations of the sinc function.

PREREQUISITES
  • Understanding of Fourier transforms, specifically the Fourier transform of rect(x).
  • Familiarity with the sinc function and its properties.
  • Knowledge of contour integration techniques in complex analysis.
  • Comprehension of Cauchy principal values and their application in improper integrals.
NEXT STEPS
  • Study the application of contour integration in Fourier analysis.
  • Explore the properties and applications of the sinc function in signal processing.
  • Learn about periodic approximations of functions and their Fourier transforms.
  • Investigate the role of Cauchy principal values in evaluating improper integrals.
USEFUL FOR

Mathematicians, physicists, and engineers working with Fourier analysis, particularly those dealing with signal processing and complex integrals.

bdforbes
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I can easily find the Fourier transform of rect(x) to be 2sinc(2\pi k) using particular conventions (irrelevant here). But when I attempt to inverse Fourier transform the sinc function, I find I have to resort to contour integration and Cauchy principal values.

This is troubling to me. It appears as if the usual definition of a Fourier transform is inadequate here, and could possibly lead to incorrect results in another context. Can anyone shed any light on this?
 
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I have come to the conclusion that the Cauchy Principal Value is the appropriate type of improper integral to use. This is because Fourier transforms originate in the consideration of periodic functions; we should consider some approximation to the sinc function which is periodic, i.e. cut the function off at some finite symmetric boundary, take the Fourier transform, and then take the limits symmetrically to infinity.
 

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