What Is the Fourier Transform?

Intuition and Mathematical Framing

• Many comments center on the idea that a signal is a linear combination of basis functions; sine waves are just a convenient orthogonal basis, not uniquely special.
• Several people emphasize the linear algebra view: Fourier transform as a change of basis in an infinite-dimensional vector (Hilbert) space; the integral kernel acts like a “continuous matrix”.
• Sines/complex exponentials are highlighted as eigenfunctions of linear, time-invariant and derivative operators, explaining why they simplify differential equations and physical models.

Band-limiting, Sampling, and Gibbs Phenomenon

• Debate over “every signal can be recreated by sines”: clarification that perfect reconstruction from samples requires band-limiting (Nyquist), but Fourier representations exist more generally, sometimes with infinitely many components and/or infinite duration.
• Distinction between band-limiting/aliasing and Gibbs ringing: commenters note Gibbs arises from rectangular windows / sinc kernels with infinite support, not from band-limiting per se.
• Short-time/windowed Fourier transforms (STFT) are discussed as the practical answer for streaming/time-local analysis, with trade-offs between time and frequency resolution.

Fourier, Quantum Mechanics, and Physics

• Position and momentum in quantum mechanics are noted as a Fourier pair; Heisenberg uncertainty is seen as a bandwidth–duration trade-off.
• Some speculative discussion links Planck scales and the universe’s finite extent to Fourier limits, flagged as “fun to think about,” not settled physics.
• More generally, many real-world systems are governed by differential equations and often oscillatory, making Fourier analysis natural and pervasive.

Fourier vs. Laplace, Wavelets, and Other Transforms

• Multiple comments argue Laplace and z-transforms are under-popularized despite being heavily used in control theory and EE; Laplace is viewed as more specialized with nicer convergence in some cases.
• Wavelets are discussed as better for non-stationary signals and certain applications, but with a narrower niche, sometimes displaced by modern ML.
• Mentions of fractional Fourier, linear canonical transforms, generating functions, and Lomb–Scargle periodograms broaden the transform “family tree”.

Applications, Compression, and Sparsity

• Uses cited include signal processing, analog electronics, control, image/audio/video compression (JPEG/DCT, MP3), OFDM, manga downscaling, color e-ink, Amazon rating heuristics, and astrophysics.
• Disagreement over “removing high frequencies doesn’t drastically change images”: one side says it produces noticeable blurring; others respond that perceptual coding mainly discards detail humans perceive weakly (especially chroma).
• Several note that many real-world signals are sparse in frequency space, which explains the power of Fourier-based compression and analysis.

Learning, Teaching, and Resources

• Strong recommendations for visual/interactive explanations: 3Blue1Brown, BetterExplained, MIT Signals & Systems lectures, explorable Fourier demos, various personal visualizations and tools.
• Some criticism that the article (and some videos) show what the FT does but not deeply why it works or how one might have invented it.
• Anecdotes from engineering education: heavy manual transform work pre-CAS, and regret from some software engineers who dismissed algebra/analysis as “useless” but later saw its relevance.