SECTION 10.3 WHEN DOES A FOURIER SERIES CONVERGE? THE FOURIER CONVERGENCE THEOREM. Suppose that f is originally deﬁned for-L ≤ x < L and the deﬁnition is then extended so that f is periodic with period 2 L . Suppose also that then f and f0 are piecewise continuous on the interval-L ≤ x ≤ L . Then the Fourier series whose coeﬃcients are given by the Euler-Fourier formulas converges to f ( x ) at points where f is continuous and to f ( x +) + f ( x-) 2 at points where f is discontinuous. Piecewise continuous means that f is continuous everywhere except for ﬁnitely many points
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