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Fourier Integral Theorem

  • Definition: Assume that f is a Lebesgue-integrable function on (-infinity,infinity). Suppose there is a point x in R and an interval [x-delta,x+delta] about x such that either a) f is of bounded variation on [x-delta,x+delta], or else b) both limits f(x+) and f(x-) exist and both Lebesgue integrals int_(0 to delta)[f(x+t)-f(x+)]/t dt and intint_(0 to delta)[f(x-t)-f(x-)]/t dt exist. Then we have the formula [f(x+)+f(x-)]/2 = (1/pi){int_(0 to infinity){int_(- infinity to infinity)[f(u)cosv(u-x)du}dv}, the integral from 0 to infinity being an improper Riemann integral.

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

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