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The theorem is that for any summability kernel $\{\phi_{n}\}$, if $f\in L_{p}(\mathbb{T}^{d})$, then $||f*\phi_{n} - f||_{p}\rightarrow 0$.

The step that I cannot follow is this:

$\left\|\frac{1}{2\pi^d}\int_{[-\pi,\pi)^{d}}(f(\cdot - t) - f(\cdot))\phi_{n}(t)dt\right\|_{p}\leq\frac{1}{2\pi^d}\int_{[-\pi,\pi)^{d}}||f(\cdot - t) - f\cdot)||_{p}|\phi_{n}(t)|dt$

I see how we could get the from the Minkowski Inequality: $\left\|\frac{1}{2\pi^d}\int_{[-\pi,\pi)^{d}}(f(\cdot - t) - f(\cdot))\phi_{n}(t)dt\right\|_{p}\leq \frac{1}{2\pi^d}\int_{[-\pi,\pi)^{d}}\|(f(\cdot - t) - f(\cdot))\phi_{n}(t)\|_{p}dt,$ which is not quite what the claim is, however.

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    You are welcome :) By the way shouldn't that be $(2\pi)^d$ instead of $2 \pi^d$?2011-10-05

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