The Uncertainty Principle for Fourier Transforms on the Real Line
نویسنده
چکیده
This paper will explore the heuristic principle that a function on the line and its Fourier transform cannot both be concentrated on small sets. We begin with the basic properties of the Fourier transform and show that a function and its Fourier transform cannot both have compact support. From there we prove the Fourier inversion theorem and use this to prove the classical uncertainty principle which shows that the spread of a function and its Fourier transform are inversely proportional. Finally, we extend our compactness result from earlier and show that a function and its Fourier transform cannot both be supported on finite sets.
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