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Tagged with bounds-of-integration analysis
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Control $\int_0^\infty |\psi(x)|^2 dx$ by $\int_0^\infty \int_0^\infty K(x+y)\psi(x) \psi(y) dxdy$
Assume that $\psi(x)$ is bounded and integrable on $x \in [0,\infty)$ with $\int_0^\infty \psi(x) dx = 0$, and suppose that $K \colon (0,\infty) \to (0,\infty)$ is some kernel function satisfying $K(x)...
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Differentiation Under the Integral Sign w Variable Substitution
Let $f\in C^1(B_\rho(\xi))$, $\xi\in\mathbb{R}^n$ and $\rho>0$. I wish to show
$$ \frac{d}{d\rho} \int_{B_\rho(\xi)} f(x)\ dx = \frac{1}{\rho} \int_{B_\rho(\xi)} nf(x) + x\cdot Df(x)\ dx $$
I'm ...
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