Fourier Theory

Differential Equations · Axiom Academy

From the orthogonality of sines and cosines to the formulas that extract every Fourier coefficient 1. Orthogonality of Trigonometric Functions The foundation of Fourier theory rests on a beautiful property: sines and cosines are orthogonal over the interval . Just as perpendicular vectors have a dot product of zero, orthogonal functions have a vanishing integral of their product. 2. Deriving the Fourier Coefficients Suppose f(x) can be written as a Fourier series: To find a_n , multiply both sides by and integrate from to . By orthogonality, every term on the right vanishes except the one matching frequency n — that term alone survives and equals . 3. Convergence of the Fourier Series A natural question: does the series actually converge back to f(x) ? The answer depends on the regularity of f . If f(x) is periodic with period and, on one period, f(x) has finitely many maxima and minima, and f(x) has finitely many discontinuities, then the Fourier series converges to f(x) at every point of continuity, and to the average of the left and right limits at each discontinuity. When a Fourier series approximates a discontinuous function, something curious happens: the partial sums overshoot near the jump by a fixed amount, no matter how many terms we include. This is the Gibbs phenomenon . For a wave jumping from -1 to +1 , the partial sums peak at about near the jump — not .

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