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# Real Fourier-Series

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The real Fourier series of a real -periodic function is an orthogonal expansion in terms of sines and cosines:

with

The type and rate of convergence of the series depends on the smoothness of . For example, sufficient for absolute convergence is that the series and converge.

A convergent Fourier series needs not to represent the true function value at every point. Usually, at a point of discontinuity, the series converges to the arithmetic mean of the left and right limits of . This explains the notation instead of , which emphasizes these convergence issues.