# Tagged Questions

Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

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### Show that a regularizing operator $K : C_c(\Omega) \to \mathcal{D}'(\Omega)$ has kernel $k \in C^\infty(\Omega \times \Omega)$.

I am reading Francois Treves' Introduction to pseudodifferential and Fourier integral operators, vol. I. Let $\Omega \subseteq \mathbb{R}^n$ be open. On page 11, Treves defines what it means for a ...
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### $W^{1,\infty}(\mathbb{R})$ is the same as $C^{0,1}(\mathbb{R})$

Let $f\in (C_c(\mathbb{R}))^*$ be a distribution. Show that $f\in C^{0,1}(\mathbb{R})$ if and only if $f\in L^\infty(\mathbb{R})$, and the distributional derivative $f'$ of $f$ also lies in ...
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### Poincaré duality for currents and non-closed forms

In page 8 of Quantization of Higher Abelian Gauge Theory in Generalized Differential Cohomology by Szabo, the author claims that Poincaré duality holds for non-closed forms as long as the other form ...
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### How to calculate the distributionderivatives of abs(x)?

Lets say we have $f(x) = |x|$. I want to calculate $f'$ and $f''$, how would I go about this? I understand that this is not defined at $x = 0$, so it will have to be done in two steps.
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### Every non-negative distribution arises from a non-negative Radon measure

A distribution on $\mathbb{R}^d$ is a continuous linear functional $\lambda: f\mapsto \langle f,\lambda\rangle$ from $C^\infty_c(\mathbb{R}^d)$ (with good seminorms topology) to $\mathbb{C}$. A ...
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### Does distribution derivative generated by $C^{\infty}$ function forces the distribution to be compactly supported?

Let $\phi \in C_C^{\infty}(\Bbb R)$. Then there exists a $\psi \in C_C^{\infty}(\Bbb R)$ such that $\psi' = \phi \iff \int_\Bbb R \phi = 0$. This is quite easy to be proved. From this it follows that ...
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### Can a “continuous” convex combination not be element of the convex hull?

Short version of question: can a "continuous" convex combination not be element of the convex hull? I am not a mathematician, so please excuse me if I am not precise. I consider first, e.g., 4 ...
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### Is the function $|x|$ in $W^{1,p}$?

I have the following question: We consider in the segment $I=]-1,1[$, the function $f(x)=|x|.$ The question is: For each value $p \in [1,+\infty[$ do we have $f \in W^{1,p}(I)$? My purpose is: We ...
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### Understanding an example of Renardy's book (example 5.48)

In the example $5.48$, authors say that: To prove that a sequence of integrable functions $f_n: \mathbb{R} \to \mathbb{R}$ converges to the delta function, it suffices to show that the primitives ...
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### Show that $u_{tt}=u_{xx}$ in the sense of Distributions

Let $u(x,t)=f(x+t)$ , where $f$ is any locally integrable function on $\mathbb{R}$. Show that $u_{tt}=u_{xx}$ in the sense of Distributions My try: For $\phi \in D(\mathbb{R})$, ...
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### Describe the distributional derivative of $f$

Let $f$ be a piece wise defined function with piece wise continuous derivative. Describe the distributional derivative of $f$. My try: If I suppose that the jump discontinuities are at the points ...
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### Distribution functions: differentials in the numerator or denominator

One paper I'm looking at says, $n(M, z) \, dM \, dz$, the number of sources with mass $M$ at a redshift $z$, in the mass interval $dM$ occurring in the redshift interval $dz$. While another ...
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### Example of a linear functional, but not a distribution

I'm looking for an example of a linear functional $u: C_c^\infty(\Omega) \to \mathbb C$ ($\Omega \subset \mathbb R^n$ open), which is not a distribution. I could not find anything... I thought of ...
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### Distributions on compact and semi-open intervals

In the theory of distributions (aka generalized functions), one considers mostly distributions $T \in \mathcal{D}(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$. Hereby, the space ...
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### Derivative of Dirac delta function as a measure

Dirac delta function can be defined in several ways. I know two definitions. One is as a distribution and the other is as a measure. I found many materials on the derivatives of delta function as a ...
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### Uniform convergence of compactly supported function

If $f\in C_c^\infty$, i.e. Compactly supported, then for $g_k(x)=f(kx)$, would $g_k\to 0$ uniformly as $k\to \infty$ since g vanishes outside a given compact set which won't be touched for $k$ ...
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