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Everything about Fourier transform

Mathematical transform that expresses a function of time as a function of frequency

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Wikipedia →

🧭 Start here: a 6-step reading path

  1. Read the overview: Fourier transformA 5-minute orientation
  2. Skim a recent paper: Application of Fourier Transform Infrared (FTIR) Spectroscopy in Characterization of Green Synthesized NanoparticlesInfluential work since 2022
  3. Pick up a classic book: A student's guide to Fourier transformsThe most-reprinted book on the subject
  4. Read the landmark paper: MAFFT: a novel method for rapid multiple sequence alignment based on fast Fourier transformCited 16,282 times
  5. Go sideways: Fourier inversion theoremA closely related topic
  6. Go sideways: Dirac delta functionA closely related topic

📖 Definitions

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🗂️ The subject at a glance

  1. Definition
  2. Background
  3. Units
  4. Properties
  5. Complex domain
  6. Fourier transform on Euclidean space
  7. Fourier transform on function spaces
  8. Generalizations
  9. Alternatives
  10. Example
  11. Applications
  12. Other notations

🏆 Landmark papers (most cited)

🆕 Influential since 2022

📚 Books worth reading

Ranked by number of editions on Open Library, a rough measure of how lasting a book is.

🔗 Related topics