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Jordan's Test

  • Definition: Assume that f is a Lebesgue-integrable function on [0,2pi] and suppose that f has period 2pi. Consider a fixed point x in [-,2[i] and a positive delta < pi. Let g(t)=[f(x+t)+f(x-t]/2 if t is in [0,delta] and let s(x)=g(0+)=lim_(t goes to 0+)[f(x+t)+f(x-t)]/2 whenever this limit exists. Note that s(x) = f(x) if f is continuous at x. If f is of bounded variation on the compact interval [x-delta,x+delta] for some delta < pi, then the limit s(x) exists and the Fourier series generated by f converges to s(x).

    Source: Mathematical Analysis, second edition by Tom M. Apostol