# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### Ways to represent functions of bounded variation as the difference of two monotone functions

I learnt (it is called Jordan's representation theorem) that every function $f : [a,b] \to \mathbb R$ with bounded variation could be represented as the difference $f = g - h$ of two monotone ...
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### Understanding last step of a proof that “two trajectories cannot cross at a finite value of t” (Phase trajectories/nodes)

Note: This proof prefaced critical points at the origin for coupled first order ODEs. It was done before showing the asymptotically stable and unstable critical points: Improper, Proper, Spiral, ...
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### Proper domain for Laplacian

it is well known that the spherical harmonics are eigenfunctions to the 3D Laplacian(angular part). But my question is: What is the right domain for this operator so that we actually get these ...
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### Formula of Squaring Sums / Integrals

I'm trying to find a proof for the identities (which I use quite often) $$\left ( \int_{a}^{\infty}f(x)\,dx \right )^2=\int_{a}^{\infty}\int_{a}^{\infty}f(x, y)\,dx\,dy$$ and similarly for the series ...
### Is $||u||_{C^\alpha} \leq ||u||_{C^1}$ for all $u$?
We have $||u||_{C^\alpha,\Omega} = \text{sup}_\Omega |u(x)|+ \text{sup}_\Omega \frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and $||u||_{C^1} =\text{sup}_\Omega |u(x)| + \text{sup}_\Omega|\frac{du}{dx}|$ I have ...
I wish to construct a function $f:\mathbb R \rightarrow \mathbb R$ of class $C^\infty (\mathbb R)$ with the folowing properties: $f(x)=0$ for $|x|\leq 1$ $f(x)=x$ for $|x| \geq 2$, $|f(x)| \leq |x|$...