{"id":"fourier-transform","domain":"applied","type":"Concept","title":"Fourier Transform","slug":"fourier-transform","url":"/mathematics/applied/fourier-transform/","summary":"Decomposes any signal into a sum of simple sine/cosine waves of different frequencies. Developed by Fourier (1807) for heat flow; now essential to signal processing, imaging, quantum mechanics.","formal_definition":"F(w) = integral of f(x)e^(-iwx)dx","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Körner, T.W. (1988). Fourier Analysis. Cambridge University Press."},{"tier":1,"citation":"Stein, E.M. and Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press."},{"tier":2,"citation":"Cooley, J.W. and Tukey, J.W. (1965). An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation, 19(90), 297-301."},{"tier":3,"citation":"Bracewell, R.N. (1999). The Fourier Transform and Its Applications. 3rd ed. McGraw-Hill."}],"relationships":[{"type":"uses","target":"integral","label":"Integral"},{"type":"related_to","target":"eulers-identity","label":"Euler's Identity — basis functions"},{"type":"uses","target":"vector-space","label":"Vector Space — L2 function space"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.7"}