Gaussian Transform

Fourier Analysis · Axiom Academy

EXAMPLE Fourier Transform of a Gaussian Computing the transform of the most important self-similar function in analysis Excellent work! You've completed this fundamental example in Fourier analysis. Here's what makes the Gaussian transform special: Self-Similarity: The Fourier transform of a Gaussian is another Gaussian. This remarkable property makes it unique among functions and central to signal processing. Completing the Square: The key technique is completing the square in the exponent, which transforms the integral into a recognizable form. Width Relationship: As the parameter a increases (narrowing the original Gaussian), the transform widens by a factor of 1/(4a) . This is a manifestation of the uncertainty principle. Gaussian Integral: The standard result is essential for evaluating the transformed function. Uncertainty Principle: The Gaussian achieves the minimum product of time-frequency spreads, making it the optimal waveform for simultaneous localization in both domains. This self-similarity property is why Gaussians appear everywhere in Fourier analysis, from the heat equation to quantum mechanics to signal processing. Practice computing transforms with different values of a to build intuition!

This is the written version of the interactive lesson above. See the full Fourier Analysis course.