Let $f$ be a complex valued Lebesgue integrable function over R, show that the function $g(t)= \int_{\mathbb R} f(x)\exp(-itx) \, dx$ is differentiable. I cannot seem to find a dominating $L^1$ function in order to apply differentiation under the integral sign. Any alternative ideas? Thanks
Why is this differentiable?
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real-analysis
analysis
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0Doesn't $|f(x)|$ do the trick? The norm of $e^{-itx}$ is 1, after all... – 2012-12-13
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0No, I require a dominating function for the derivative of the integrand, or for the difference f(x){exp(-itx)-exp(-isx)}/(t-s), I don't know how else to approach this problem. – 2012-12-13