Use this tag for questions about Schwartz distributions, also known as Generalised Functions. For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).
4
votes
1answer
110 views
Generalizing the weak derivative
I am wondering about the weak derivative in time. We say f has a weak derivative f' if $$\int_0^T f\phi' = -\int_0^T f'\phi$$
for all $\phi \in C_0^\infty(0,T)$.
This definition uses the $L^2$ inner ...
1
vote
1answer
64 views
How to prove the density result？
How to prove that $C_c^\infty(R^n)$ is dense in $C^0(R^n)$,where the topology of $C^0(R^n)$ defined as follows
$u_k\rightarrow 0$ in $C^0(R^n)$ if and only if $\forall K\subset\subset R^n$,$sup_{x\in ...
-1
votes
1answer
113 views
Fundamental solutions of PDEs
I have two questions about solving PDEs. $L$ is an linear differntial operator
In the complement of the origin, the equation $LE =\delta$ reduces to $LE = 0$. Why?
What can say about solutions of $ ...
-1
votes
1answer
60 views
When Dirac function is in $H^{-m}(R^n)$?
If Dirac function $\delta\in H^{-m}(R^n)$,please give the range of $m$?
9
votes
1answer
125 views
How much can a weak derivative differ from a classical one?
Let $B$ denote the unit ball in $\mathbb{R}^n$ and let $f\in C^1(B\setminus\{0\})\cap L^1(B)$. Denote with $\nabla_c f$ the classical gradient, which is defined in $B\setminus\{0\}$, and denote with ...
6
votes
1answer
109 views
Convolution between two distributions
I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say:
$$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$
where ...
3
votes
1answer
63 views
Equicontinuity and uniform boundedness for “distributions”
Exercise. (Rudin, Functional Analysis, chapter 2, pag. 53). Let us consider the space
$$
\mathcal D :=\{f \in C^{\infty}(\mathbb R), \, \text{supp}f\subseteq [-1,1] \}
$$
with the topology induced by ...
2
votes
0answers
67 views
Properties of Fourier transform of distributions
For distributions the scaling property, $f(ax) = \frac{1}{|a|} \mathcal{F(\frac{u}{a})}$, of the Fourier transform is no longer true. Is there a source that lists which properties of the Fourier ...
2
votes
0answers
45 views
Applications for the tail bounds for the hypergeometric distribution
I am interested in the applications for the probability tail bounds (non-asymptotic) for the hypergeometric distribution. I would be very appreciative if you can name or link me to some.
1
vote
1answer
63 views
Tail bound for hypergeometric distribution
I am looking for a reference (book) for the tail bound for the Hypergeometric distribution.
I know there is a nice paper by Skala (2009) but its unpublished. I am looking for a book which would be a ...
1
vote
1answer
93 views
weak derivative and continuous functions (functionals, distributions)
Suppose $\varphi \in C_c^\infty(0,T; H^1(\Omega))$ is a $H^1(\Omega)$-valued test function (it vanishes at $t=0$ and $t= T$), and $f \in C^1(0,T \times \Omega)$. Let $w \in L^2(0,T;H^1(\Omega)$ with ...
1
vote
0answers
35 views
Understanding analyticity
Assume $\omega , \mu \in \mathcal{D}'(\mathbb{R})$ are distributions with $\operatorname{supp}\mu $ compact, that are related according to
$$
\omega = \varphi \ast \mu = \int (x-y)^{1/2}_+ \mu (y) \, ...
1
vote
0answers
19 views
Function of the incremental ratio tends weakly to a distribution
Let $g:\mathbb{R}^3\to\mathbb{R^2}$ be a continuous function. Suppose that there exists $\Omega$ a neighborhood of $0$ where $Xg, Yg \in L^\infty(\Omega)$, with
...
3
votes
1answer
65 views
Show continuity of a function?
Are there theorems or results to show that if for every $\varphi\in \mathcal{C}_0^\infty(\mathbb{R})$ we have,
$$\int_{\mathbb{R}} \varphi^k(x)\mu(dx) \leq C$$
Then $\mu(dx) = f(x)dx$ and $f\in ...
0
votes
1answer
93 views
Questions related to distribution function and its “inverse”
Let $f: \mathbb R^n \to \mathbb R$ be a measurable fucntion. Define $F(t) = \mu \{x:|f(x)| >t\}$
Show that $F$ is nonincreasing and right-continuous (done).
Define $F^\star(v)=\inf \{t: F(t)\leq ...
2
votes
1answer
45 views
A particular product of distributions
Suppose you have two continuous functions $f,g: \mathbb{R}\to\mathbb{R}$; is the product $f'g$ as a distribution, at least locally? I am interested in a local result, actually, so you can as well ...
0
votes
1answer
79 views
Composition of a distribution with a map
Suppose $\lambda \in C_c^\infty(\mathbb{R})^*$ is a distribution and $f: \mathbb{R} \to \mathbb{R}$ is a continuous map of the real line. In addition assume $f$ has compact support. How can I make ...
1
vote
1answer
63 views
The Dirac impulse and Fourier transform
Here wikipedia it is said that the Dirac delta could be thought of as
$$
\delta(x) = \left\{ \begin{array}{ll} \infty &, x = 0 \\ 0 &, x \ne 0 \end{array}\right.
$$
and here that the ...
3
votes
0answers
272 views
Property of derivative of Dirac delta function in $\mathbb{R}^n$
With reference to Property of Dirac delta function in $\mathbb{R}^n$, is there a similar formula for $\langle g^*\delta', f \rangle$ (or even $\langle g^*\delta^{(n)}, f \rangle$)? By similar I mean a ...
1
vote
1answer
92 views
Distributional/weak time derivative basic question
Suppose we have $u \in L^2(0,T;H^1(\Omega))$, and $v \in L^2(0,T;H^{-1}(\Omega))$ is the weak time derivative of $u$, so by definition it satisfies
$$\int_0^T u(t)\phi'(t) = -\int_0^T v(t)\phi(t)$$
...
4
votes
1answer
120 views
Convolution square root of $\delta $
I want to somehow classify the distributional solutions of the equation
$$
f \ast f = \delta
$$
where $\delta = \delta _0$ is the Dirac delta distribution. Clearly, by Fourier transformation, we ...
2
votes
0answers
109 views
Proof that limit goes to zero without Riemann-Lebesgue lemma
Let $\varphi$ be a test function ($\varphi$ is smooth and has compact support - is zero outside some bounded interval).
I know that the following
$$ \lim_{\epsilon \to 0_+} ...
1
vote
0answers
45 views
Lebesgue-Stieltjes integral as a generalized function
Given some convex function $f(x)$, $x >0$ we can define a distribution $F \in \mathcal{D}'(0,\infty)$ using Lebesgue-Stieltjes integral
$$
\langle F, \varphi \rangle ...
3
votes
1answer
112 views
Second derivative of convex function
Let $f(x)$, $x>0$ be a convex function. Then it's distributional second derivative is defined by the rule
$$
\langle f''(x),\varphi(x)\rangle = \langle f(x), \varphi''(x)\rangle
$$
for any ...
1
vote
1answer
67 views
Sobolev spaces of infinite order
I do have a question about the Sobolev spaces of infinite order. Let me first define them:
Let $H^s(\mathbb{R}^n)$ denote the Sobolev space of order $s \in \mathbb{R}$. We can naturally identify ...
1
vote
1answer
154 views
Principal value of 1/x- equivalence of two definitions
As far as I know, the principal value of a non-summable function like $1/x$, denoted $\mathcal{P}(1/x)$, is a distribution that that acts on some smooth function $f$ in some test-function space and ...
2
votes
2answers
82 views
particular solution by variation of constants
I have this ODE : $y^{''}(x)-Ay(x)=Bx \delta_{0}(x)$ where $A,B$ are constants and $\begin{equation}
\delta_{0}=
\begin{cases}
\infty & \text{if $x=0$}, \\
0& \text{else}.
\end{cases}
...
1
vote
1answer
99 views
How to show that limit is a delta function
Let $\{\phi_{n}(t)\}_{n=1}^{\infty}$ be a complete orthonormal system at $[a,b]$. Then
$$
\sum\limits_{n=1}^{\infty} \phi_{n}(t)\phi_{n}(s) = \lim\limits_{N \to \infty} \sum\limits_{n=1}^{N} ...
1
vote
1answer
96 views
Singular support of distributions
I've read the following definition:
Let $u \in \mathcal{D}^\prime(\mathbb{R}^n)$. We say that $y_0 \notin \mathrm{sing} \ \mathrm{supp} \ u$ if there exists $\phi \in ...
1
vote
1answer
82 views
Extension of formula for solution of heat equation on distributions
Consider a heat equation
$$
u_{t} = u_{xx}+f(t,x),\; (x,t) \in (0,L)\times(0,T] \\
u(0,x) = 0, \; x \in [0,L] \\
u(t,0) = 0, \; t \in [0,T] \\
u(t,L) = 0, \; t \in [0,T].
$$
If $f(t,x)$ is ...
2
votes
0answers
53 views
Does multiplication commute with taking of fundamental solution (heat equation)
Let $\Phi(t,x)$ be a heat function,
$$
\Phi(t,x) = \frac{1}{\sqrt{4 \pi t}} \exp\left(-\frac{x^2}{4t}\right).
$$
Then $(\partial_{t} - \partial_{xx})\Phi(t,x) = \delta(t)\delta(x)$. Furthermore,
...
2
votes
0answers
42 views
Representation of distribution by nonnegative measure
Let $T \in \mathcal{D}'(\mathbb{R}_{+})$ be a distribution on $\mathbb{R}_{+}$ such that for any $f \in \mathcal{D}(\mathbb{R}_{+})$, $f \geqslant 0$ we have
$$
\langle f, T \rangle \geqslant 0
$$
...
2
votes
0answers
56 views
Weak derivative and homeomorphisms commute
Suppose $\phi:H^1(S) \to H^2(R)$ is a linear homeomorphism, where $S$ and $R$ are compact surfaces in $\mathbb{R}^n$.
Let $\varphi \in D(0,T;H^1(S))$ be a $H^1(S)$-valued $C_c^\infty(0,T)$ function. ...
1
vote
1answer
54 views
Convergence of distributions in $L^p$
If I understand correctly, distributions $F_n \in C^\infty_c(\mathbb{R})^*$ are defined based on how they act on test functions $\phi \in C^\infty_c(\mathbb{R})$.
What does it mean then to say $F_n ...
1
vote
0answers
19 views
Solve an equality for distribution
I have an equality that holds for any $\lambda > 0$
$$
\int\limits_{0}^{\infty}{e^{-\lambda t^{\alpha}}} T(t)dt = \int\limits_{0}^{\infty}e^{-\lambda t^{\alpha}} dg(t),
$$
where $\alpha > 0$ ...
4
votes
4answers
649 views
On the derivative of a Heaviside step function being proportional to the Dirac delta function
I am learning Quantum Mechanics, and came across this fact that the derivative of a Heaviside unit step function is Dirac delta function. I understand this intuitively, since the Heaviside unit step ...
2
votes
0answers
412 views
Distributions supported on a single point
Let $d=1$.
(i) Show that if $\lambda$ is a distribution and $n\geq1$ is an integer, then $\lambda x^n=0$ if and only if $\lambda$ is a linear combination of $\delta:=\delta_{\{0\}}$ and its first ...
3
votes
1answer
77 views
What is the use of $H_s$ for non-integer $s$?
So we have the whole set of theory for Sobolev spaces \begin{equation}
H_s(\mathbb{R}^d)=\{u\in D'(\mathbb{R}^d):(1+|y|^2)^{s/2}\hat{u}\in\mathcal{L}^2(\mathbb{R}^d)\},
\end{equation} and we know that ...
2
votes
1answer
48 views
Non-regularity of non-elliptic operator
Let $\Omega\subset\mathbb{R}^d$ be open,and $P(D)=\operatorname{\sum_{|\alpha|\le N}}f_{\alpha}D^{\alpha}$ be an elliptic differential operator. Rudin proves in Functional Analysis Part II the ...
3
votes
1answer
109 views
Show the usual Schwartz semi-norm is a norm on the Schwartz space
Let $f \in C^\infty(\mathbb R)$. Define the semi-norm
$$
\|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)|
$$
where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$.
Show ...
4
votes
1answer
82 views
How to prove that $x \rightarrow e^{1/x}$ is not a restriction of any real distribution to $ \mathbb {R}_+$?
This is an excercise 2.2 from Hormander, vol. I:
Does there exist a distribution $u$ on $\mathbb{R}$ with the restriction
$x \rightarrow e^{1/x}$ to $\mathbb{R}_+$?
The answer, provided in the book, ...
3
votes
1answer
91 views
How to understand limit
Let $\chi(t)$ be the Heaviside function, i.e. $\chi(t) = 1$ for $t > 0$ and $\chi(t) = 0$ if $t \leq 0$. Reading a paper I faced with a statement that
$$
\frac{t^{p-1}}{\Gamma(p)}\chi(t) \to ...
2
votes
1answer
126 views
Schwartz space: semi norm estimate on translation
the following family of semi norms is commonly used to introduce the space of Schwartz functions $\mathcal{S}(\mathbb{R}^n)$:
$$
\|\phi\|_N := \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha|\,,|\beta| ...
0
votes
1answer
80 views
Fundamental solution of a particular differential operator
Show that he distribution given by the locally integrable function $\dfrac{1}{2} e^{|x|}$ is a fundamental solution of the differential operator
$\begin{equation}
-\dfrac{\partial^{2}}{\partial ...
1
vote
1answer
61 views
Proving a sequence of distributions converge in $C^{-\infty}(\mathbb{R})$
Question: I have a sequence $(T_n)$, where $T_n$ is given by the locally integrable function $ne^{-\dfrac{n^{2}x^{2}}{2}}$, converges in $C^{-\infty}(\mathbb{R})$ and compute its limit.
I suspect ...
2
votes
1answer
43 views
So $k^2-\Delta: H_{s+2}\to H_{s}$ is a homeomorphism, but what does that tell us?
For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation}
H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\},
\end{equation} where $\mathcal{S}'$ is the space of ...
3
votes
1answer
267 views
Are smooth functions with compact support weakly-* dense in $L^\infty$?
My question is this : given $f \in L^\infty(\mathbb{R}^2)$, can we find a sequence $\phi_n$ of smooth, compactly supported functions (test functions) such that for any $g \in L^1(\mathbb{R}^2)$,
...
1
vote
1answer
38 views
Easy question about Bochner space
Question Suppose $u$ and $v$ are in $L^1(0,T; X)$ where $X$ is Banach. Suppose v = u' in the distributional sense. I want to show that, for $w \in X^*$, that $$\frac{d}{dt}\langle w, u \rangle = ...
0
votes
1answer
73 views
How does a myopic interpret Wiener's Tauberian?
I just read about this post on the intuition behind convolution. In Terence Tao's answer convolution is interpreted as the blur of image in near-sighted eyes. In Harald Hanche-Olsen's it is made ...
1
vote
1answer
50 views
Singular support of a tempered distribution is compact?
I am reading Introduction to the Theory of Distributions by Friedlander and Joshi. As definition 8.6.1, they define the singular support of a tempered distribution $u$ to be the complement of {$x$: ...
