Fourier Coefficients

Real Analysis · Axiom Academy

How orthogonality turns a wave into numbers — each coefficient is the projection of f onto one sine or cosine. 1. A Function Built from Waves A -periodic function f can be written as a constant plus a sum of cosines and sines at every integer frequency. The numbers a_0,a_n,b_n say how much of each wave is present. A constant term, plus one cosine and one sine at each frequency n 2. Orthogonality: the Waves Don't Overlap Over , multiply any two different members of the family and integrate — you get exactly zero. Multiply a member by itself and you get . This is the whole engine of the derivation. The signed area of the product for cancels perfectly: every positive lobe is matched by a negative one. The functions are orthogonal — the continuous analog of a zero dot product. 3. Finding a_0 : the Constant Term Integrate the whole series from to . By orthogonality every sine integrates to 0 and every cosine ( ) integrates to 0 — only the constant survives. Every and term integrates to 0 over the full period, leaving only the constant: To isolate a_n , multiply the series by and integrate. Orthogonality kills every term except the one cosine at the matching frequency — the projection of f onto that single wave. Only the n=m term survives the orthogonality test: 5. Finding b_n , and the Symmetry Shortcut The same move with gives the sine coefficients. And symmetry can hand you half the answer for free: project an even f onto a sine and the area always cancels to zero.

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