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).

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2
votes
2answers
855 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
99 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
148 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
379 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 ...
2
votes
1answer
158 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
74 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
70 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
90 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
109 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
30 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 $\alpha > 0$ ...
5
votes
4answers
3k 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
677 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
121 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
69 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
483 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
102 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
102 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 ...
3
votes
1answer
280 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
245 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
99 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
57 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 ...
4
votes
1answer
599 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
2answers
68 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 = ...
1
vote
1answer
104 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
172 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 {$x$: ...
1
vote
1answer
191 views

Weak derivative of $\operatorname{sgn}(x_1)$

Let $x\in \mathbb{R}^{n}, x = (x_1,\ldots,x_n)$, and $f(x) = \operatorname{sgn}(x_{1})$. Is $f$ weakly differentiable on $U = B(0,1)$, i.e. unit ball in $\mathbb{R}^{n}$, and what is the weak ...
2
votes
2answers
65 views

Convergence of a sequence in $L^1(\mathbb{R}^3)$

All function spaces are over $\mathbb{R}^3$. Let $u_n \in C^\infty_0$, $u_n\rightarrow u$ in $L^1$. Let $v\in L^1_\text{loc}$ be such that $uv \in L^1$. Does $u_n v \rightarrow uv$ in $L^1$? What ...
2
votes
4answers
389 views

What is the appropriate topology on $C_c^\infty (\mathbb{R}^d)$?

Let $\{ U_k:k\in \mathbb{N}\}$ be an increasing sequence of open subsets of $\mathbb{R}^d$ whose union is $\mathbb{R}^d$ and such that each $K_k:=\overline{U_k}$ is compact and $K_k\subseteq U_{k+1}$. ...
1
vote
1answer
68 views

Identify the distrionbutional derivative with classical derivative?

I am reading Rudin's Functional Analysis and got quite confused by his proof for theorem 7.25, which he calls Sobolev's Lemma. In proving the theorem, he defines the function $F$, and calculates its ...
1
vote
1answer
491 views

Easy question on derivative in the sense of distribution

I would like help proving this elementary result: Let $f\in L^{1}_{loc}(a,b)$. Let $x_0 \in (a,b)$ Let $F(x)=\int^{x}_{x_0} f$. Then $F'=f$ in the sense of distributions. i.e How do I show $\langle ...
4
votes
2answers
186 views

Howto show that function is a representation fot the delta function via complex path integrals?

So given is the definition: $$ f(x):=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}e^{ikx}dk $$ I'm supposed to show that this is a representation of the Dirac delta "function" ($f(x) = \delta(x)$) ...
1
vote
1answer
401 views

Delta Dirac Function

Show that $\delta$, function of Dirac, defined than $\left<{\delta_0,\phi}\right> = \phi(0)$ belongs to $W^{-1,p}(]-1,1[)$ and $\delta_0 \notin L^p(-1,1)$ $\forall p\geq 1$. How I will be able ...
1
vote
2answers
179 views

Why begin with distributions and then move to tempered ones?

After reading several books on distribution theory, I got a strange feeling. Why do they all begin with the theory of distributions and then move on to tempered distributions? Why can't we just start ...
6
votes
2answers
330 views

Paley-Wiener type theorems for distributions?

In general a theorem of Paley-Wiener type gives a relation between the decay of a function and the smoothness of its Fourier transformation, and there are plenty of them since there are many kinds of ...
1
vote
1answer
151 views

If $f$ is a bounded tempered distribution and $g \in L^1$ is then $\int_{\Bbb R^n}(f\ast\tilde\varphi)(x)\tilde g(x)\,dx$ a tempered distribution?

Let $f$ be a bounded tempered distribution, that is, $f\ast\varphi \in L^\infty(\mathbb R^n) $ for every Schwartz function $\varphi$. If $g \in L^1(\mathbb R^n)$, does the following definition define ...
2
votes
1answer
72 views

Laplacian in $\Bbb R^2$ acting on compact test-function

I am trying to follow an argument in Strichartz's "A Guide to Distribution Theory and Fourier Transforms" We consider $\langle \Delta u, \rho \rangle$ where $\Delta u$ is the two dimensional ...
2
votes
1answer
343 views

What is good about homogeneous functions?

Given $r>0$ and $f:\mathbb{R}^n\to \mathbb{R}$, $d_rf$ is the function defined by \begin{equation}d_rf(x_1,x_2,\dots,x_n)=f(rx_1,rx_2,\dots,rx_n)\end{equation} and is called the $r$-dilation of ...
0
votes
2answers
170 views

How to cook up test functions?

Let $\Omega\subset \mathbb{R}$ be open. A test function is a $C^\infty$ function with compact support. This is a rather strong restriction, for instance, no analytic function is a test function. But ...
1
vote
0answers
86 views

What is $\overline{\partial} 1/z^2$?

it is all in the title : what is $\overline{\partial} \frac{1}{z^2}$ in the sense of distributions ? I remember that $\overline{\partial} \frac{1}{z}$ is a dirac at 0, but I can't seem to find a way ...
5
votes
2answers
384 views

Fractional derivatives of delta function $ \delta (x) $

How can I define the fractional derivative of the Delta function? I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) ...
2
votes
1answer
149 views

generalized functions (Distributions) elementary question

I am working with Strichartz's "A Guide to Distribution Theory and Fourier Transforms" (self-study -> not a homework question). He says none of the distributions that correspond to $1/|x|$ are ...
0
votes
1answer
119 views

Confused by a proof in Rudin's Functional Analysis

I am referring to a proof in Part II of Rudin's Functional Analysis. I got confused by his proof of Thm 6.26 (page 167). He says by applying (2) successively we can get inequality (4), but I do not ...
1
vote
1answer
61 views

Nice example where $D^{\alpha}\Lambda_{f}\neq\Lambda_{D^{\alpha}f}$?

Let $\Omega\subset\mathbb{R}^n$ be open and $f$ be a locally integrable function. The distribution associated with $f$, $\Lambda_{f}\in D'(\Omega)$, is defined via \begin{equation} ...
2
votes
0answers
70 views

Good references on Distribution Theory [duplicate]

Possible Duplicate: Distribution theory book Two books I have been reading are Strichartz's A Guide to Distribution Theory and Fourier Transforms and PartII of Rudin's Functional Analysis . ...
2
votes
0answers
99 views

Liouville's Theorem in $\mathbb{R}^n$

Liouville's Theorem states that if a tempered distribution is harmonic, $\Delta{u}=0$, then $u$ is given by a polynomial. For the argument, we take Fourier transform of $\Delta{u}=0$ to obtain ...
4
votes
2answers
514 views

Convergence of test-functions is not induced by any metric.

By $\mathcal{D}(\mathbb{R})$ we denote linear space of smooth compactly supported functions. We say that $\{\varphi_n:n\in\mathbb{N}\}\subset\mathcal{D}(\mathbb{R})$ converges to ...
1
vote
1answer
111 views

Delta function question

Given the functions $$f(x)= \delta (x-a)$$ $$g(x)= \frac{1}{a} \delta \left(x- \frac{1}{a}\right)$$ for a real constant $a\gt0$, is there a relationship between $f$ and $g$? I believe that $ ...
5
votes
2answers
394 views

Square root of compactly supported C-infinity function

Given $u \in \mathcal{C}^\infty_0(\mathbb{R}^n)$, $u \geq 0$ everywhere, is $v(x) = \sqrt{u(x)}$ also in $\mathcal{C}^\infty_0$? It is clear that the only problematic points are the boundary of the ...
2
votes
2answers
240 views

Regarding the definition of Schwartz Space of functions

I came across a definition of Schwartz Space where they were defined as functions $f$ such that $\mathrm{lim}_{|x|\to \infty} |x^{\alpha}D^{\beta}f(x)|=0$ for any pair of multiindices $\alpha,\ ...
0
votes
1answer
164 views

how to compute the convolution of two measures explicitly

Here is my example:u and v are the surface measures on the spheres {${x;|x|=a}$} and {${x;|x|=b}$} in $\mathbb{R}^{3}$.Then what's $u\ast v$ ? And what if in $\mathbb{R}^{n}$?