{"id":"fourier-series","domain":"analysis","type":"Concept","title":"Fourier Series","slug":"fourier-series","url":"/mathematics/analysis/fourier-series/","summary":"Expresses periodic functions as sums of sine/cosine waves. Fourier (1807), initially met with skepticism. Underlies MP3/JPEG compression and the Fast Fourier Transform algorithm.","added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Fourier, J. (1822). Théorie analytique de la chaleur. Firmin Didot."},{"tier":1,"citation":"Carleson, L. (1966). On convergence and growth of partial sums of Fourier series. Acta Mathematica, 116, 135-157."},{"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":"Körner, T.W. (1988). Fourier Analysis."}],"relationships":[{"type":"generalised_by","target":"fourier-transform","label":"Fourier Transform"},{"type":"related_to","target":"taylor-series","label":"Taylor Series"},{"type":"motivated_by","target":"differential-equation","label":"Differential Equation — heat equation"},{"type":"uses","target":"big-o-notation","label":"Big-O Notation — FFT efficiency"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"2.4"}