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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53 views

Approximate dirac delta and integration error

For a sequence of functions $g_n(x-x_o)$ approximating the Dirac delta I can write: $ \int_a^b g_n(x-x_o) f(x) dx = \int_a^b \delta(x-x_o) f(x) dx + \epsilon_n$ when $x_o \in [a,b]$. I am trying to ...
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2answers
77 views

Distributional derivatives on hypersurface?

In a paper I was reading, the define a set $Q=(0,T)\times \Omega$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain, and then they write $$\langle \frac{d}{dt}u - \Delta u, \varphi ...
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0answers
37 views

Convolution of two delta distributions

Show ${\int}_0^{\infty}\delta(x+z)\delta(y-z)dz =\delta(y+x)$ It seems obvious, and I don't think we need a rigorous proof for this (statistical mechanics homework) but I want a rigorous proof of ...
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1answer
58 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
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2answers
63 views

Delta function multiplied by an exponential function

I do not know if this is an ill-posed question but ... is $\delta(t)e^{-\gamma t}$ equal to $\delta(t)$? Thanks, biologist
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1answer
61 views

Applications of the theory of distributions outside of PDEs?

Are there any interesting, important or powerful mathematical applications to the Theory of Distributions besides those dealing with partial differential equations?
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1answer
40 views

Weighted Dirac comb as a tempered distribution?

I'm trying to determine when a "weighted" Dirac comb is a tempered distribution. More precisely, trying to prove: $$u=\sum_{k=1}^{\infty}c_k \delta_k\in\mathcal{S}'(\mathbb{R})\iff\exists ...
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2answers
43 views

A short question concerning the distributional solution of $xf=0$

I was reading my notes on the following result: All the $\mathcal{D}'(\mathbb{R})$ solutions to $xf =0$ are of the form $c\delta $ where $c$ is constant and $\delta$ is the dirac delta distribution ...
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1answer
55 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
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1answer
38 views

$(1+x^2)T = 1$ in $\mathscr D'(\mathbb{R})$

I only know that the solutions of $xT=0$ in $D'(\mathbb{R})$ are in the form $c\delta_0$, but I can't figure out how to find general solution to $(1+x^2)T = 1$. Any ideas?
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22 views

distribution sense

We have, \begin{eqnarray*} \Psi=\int_0^L \left[\frac{1}{2}\left((1+a^2)u_x^2+2a u_x |u_x|\right)+F(u)+\frac{1}{2}w^2\right]\,dx \end{eqnarray*} We can write, formally, \begin{eqnarray*} \delta ...
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1answer
47 views

How to prove that limit is equal to zero

How to prove that: $$\lim_{\epsilon\rightarrow 0}(-\log(-x) \phi(x)|_{-\infty}^{-\epsilon} -\log(x) \phi(x)|_{\epsilon}^{+\infty})$$ where $\phi(x) $ is any test function is equal to $0$. It seems ...
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4answers
111 views

Why is $C_c^\infty(\Omega)$ not a normed space?

I am watching a Coursera video on Théorie des Distributions and I am trying to understand one of the slides. Let $\Omega \subset \mathbb{R}^N$ be an open set and $C_K^\infty(\Omega) = \{ \phi \in ...
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1answer
47 views

Relationships between growth rates of a distribution and smoothness of its Fourier transform

Let $f\in \mathcal{S}^\prime(\mathbb{R})$ be a tempered distribution, and $\hat{f}$ be its Fourier transform. It is known that when both $f$ and $\hat{f}$ are $L^2$ functions, there are relationships ...
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1answer
42 views

About a condition for a distribution to be zero

I'm trying to solve the following question: If $S$ is a distribution with compact support on $\mathbb{R}$, which verifies $\langle S, x^n \rangle=0$ $\forall n\in\mathbb{N}$, then $S$ is the ...
3
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2answers
78 views

Fundamental solution of a differential operator

I'm trying to solve this question, but I don't know how to deal with it: If we have $b=(b_1,,\dots,b_n)\in\mathbb{R}^n$ and $\beta\in\mathbb{R}$, prove that the differential operator ...
2
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1answer
85 views

Fourier transforms of the Heavyside function and the absolute value function

I'm trying to obtain these two Fourier transforms. First of all, I'm using the following definition of Fourier transform: $$\cal{F}(f)(y)=\int_{-\infty}^{\infty}f(x)e^{-ixy}\;dx$$ What I have so ...
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0answers
30 views

Mellin-Barnes transform of $\frac{1}{\Gamma}$

Let $\varphi \in L_1(c+i\mathbb R)$, where $c > 0$. Then we can define the Mellin-Barnes transform (or the inverse Mellin transform) of the function $\varphi$ by the formula $$ \mathcal ...
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1answer
51 views

Fourier transform of $\frac{d}{dt}\ln\frac{1}{it}$

I'd like to proove the identity $$\mathcal{F}\left(\frac{d}{dt}\ln\frac{1}{it}\right)=2i\pi H$$ with $H=\mathbb{I}_{\mathbb{R}^+}$ ie the Heaviside step function, $\mathcal{F}$ denote the Fourier ...
0
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1answer
72 views

Schwartz Space: I need to prove that $f \in \mathscr S'(\mathbb R)$

I need to prove that $f\in \mathscr S'(\mathbb R)$, where \begin{equation} f(x)= \left\{ \begin{array}{cl} 0, & \mbox{$x \leq 0$}\\ x, & \mbox{$x>0$}\end{array} \right. ...
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0answers
32 views

If integral equals zero, function equals zero almost everywhere [duplicate]

Let $Int\Omega=\Omega\subset\mathbb{R}^n$, $u\in L_{loc}^1(\Omega)$ and $u$ is continous. Prove that: $\int_\Omega u(x)\phi(x)dx=0$ for any $\phi\in D(\Omega)$ then $u=0$ almost everywhere. ...
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0answers
58 views

Does the norm exist on $\mathscr S' \subset \mathscr D'$?

I take an extension from $L^2(\mathbb R)$ to $\mathscr S'(\mathbb R)$ of tempered distributions for a mapping of nonlinear distribution. I do not want to use seminorm, but the norm, therefore the ...
2
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2answers
53 views

$xT' = 1$ in $D'(\mathbb{R})$

I need help solving the following problem: I want to show that all solutions of $$xT' = 1\ , T \in D'(\mathbb{R})$$ take the following form: $c_{1} + c_{2}1_{[0, \infty)} + ln|.|$ What I tried so far ...
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1answer
111 views

Extend from the spaces $L^2$ to the tempered distributions $\mathscr S'$

Recall that $\mathscr S(\mathbb R) \subset L^2(\mathbb R)$. Assume the continous and bounded functions $f: \mathbb R \to \mathbb C, \,f \in C(\mathbb R), f(t) \not= 0$ only if $a \leq t \leq b$ and ...
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2answers
70 views

Convolution and Dirac delta [closed]

I need to prove the following: $\delta_0 * \phi = \phi$, where $\phi$ is a test function. Thank you for your help.
2
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0answers
32 views

What is the motivation for “continuity in the sense of distributions”?

Let $M$ be a compact (real) manifold and let $\Omega^m_c(M)$ be the compactly supported $m$-forms on $M$. Apparently a linear map $T : \Omega^m_c(M) \to \mathbb{R}$ is continuous "in the sense of ...
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1answer
53 views

How to solve distributional equation?

What are the only solutions of a distributional equation: $$xT'=0$$ Thanks. Any hint? I know that $T'(\phi)=-T(\Phi')$.
2
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1answer
36 views

Translation is continuous

Let $\mathcal D$ be the space of 'test-functions'. Those are infinitely differentiable functions with compact support. Define the following convergence on $\mathcal D$. $(\phi_j) \to \phi$ in ...
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0answers
15 views

What is the modulus of smoothness for Wigner-Ville Distribution?

I heard today a seminar where the speaker talked about General Shannon Sampling operators and its modulus of smoothness. I can only find this article about the modulus of smoothness. I think you can ...
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16 views

Boundary of real part of functions in $H^p$ and Poisson nontangential maximal function

I have two questions when reading on $H^p$ spaces, many books do not give their proofs. First we reminde that $H^p(\mathbb R^2_+)$ consists of all functions $F$ which is analytic in the upper half ...
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2answers
97 views

Fourier transform of a unity function and of unit step function

Fourier transform of the unity function is the Dirac delta distribution. I think this means: In particular, the Fourier transform of the unity function is the Dirac delta distribution, ...
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1answer
51 views

Is delta distribution continuous and differentiable with dual space norm?

I know that delta distribution $\delta : \mathcal S (\mathbf R) \to \mathbf C$ is continuous with usual seminorm and here. I am interested in its continuity with dual-space $H^{-1}(\Omega)$ of ...
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0answers
34 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
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1answer
55 views

What is the interpretation of $\delta(x)\ln\delta(x)$, where $\delta(x)$ is the Dirac's delta function?

What's the result of the following integral? $$\int f(x)\delta(x)\ln\delta(x)\mathrm{d}x$$ where $f(x)$ is a smooth function (continuous derivatives of as high order as needed).
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1answer
54 views

When is $W^{m,p}(\Omega)$ dense in $L^p(\Omega)$?

Let $\Omega\subset\mathbb{R}$, $m\in\mathbb{N}$ and $1\leq p<\infty$. What are the (most common) sufficient conditions (if such conditions exists) that we can impose on $\Omega$, $m$ and $p$ to ...
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1answer
36 views

Distribution of a product of RVs

I have this question which I cannot figure out where I was doing it wrong. Let $(X,Y)$ be a jointly continuous RV with density function $$f_{(X,Y)}(x,y)=\frac{12.5}{LW}\;\;\text{for}\; \;0.9L\leq ...
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1answer
93 views

Fundamental solution of nonlinear PDE

A fundamental solution of a linear PDE (in sense of Schwartz), $Lu=0$ is defined as a distribution $E$ such that $LE=\delta$. Now I wish to find fundamental solution of nonlinear PDE, such as the ...
2
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2answers
75 views

Computation of integral involving Heaviside function

Let $H : \mathbb{R} \to \mathbb{R}$ denote the Heaviside function: $$ H(y) = \begin{cases} 0 & y < 0, \\ 1 & y \ge 0. \end{cases} $$ Suppose that $c > 1$ is fixed with $t$ ...
0
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1answer
28 views

Quadratic Time-Frequency Representation with L2 norm

I have been reading the book Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition. They use different norm for different problems in their automatic ECG detection ...
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1answer
69 views

Contraction map in extending domain from Dense subset to $L^{2}$

This thread is about extending a dense domain $D \subset L^{2}$ into $L^{2}$. I do not understand what Deyton means in his comment about getting contraction map when doing this. I cannot see any ...
6
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4answers
164 views

Dirac delta of nonlinear multivariable arguments

How does one compute a dirac delta function with a multivariable argument? For example, compute: $$ \int^{\infty}_{-\infty}{\rm d}x\,{\rm d}y\, \delta\left(x^{2} + y^{2} - 4\right) ...
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1answer
248 views

integral of Dirac delta function with sine

It i well known that the Dirac Delta Function has the following property $\int_{-\infty}^{\infty}\delta(t-a)f(t)dt=f(a)$ if $g(t)=\int_{0}^{t}\sin(t-\tau)\delta(\tau-\pi)d\tau$ then $g(t) = ...
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1answer
68 views

Is $\delta : \mathcal{S}(\mathbf{R}) \to \mathbf{C}$ continuous with usual seminorm?

I am thinking again the accepted answer which is found here: When viewing $\delta: \mathbf{S} \to \mathbf{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space ...
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2answers
30 views

To write this sentence about a distribution more rigorously

I have the sentence at the moment Notice that for all $c \in \mathbf{C}$ such that $W(cx) = |c|^{2} Wx$. which I do not like. I mean to say that for all $c \in \mathbf C$ the equation is true. ...
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1answer
57 views

Are these $L^{p,q}$ Lebesgue spaces?

I have a theorem, deduced mostly from Loukas' Classical Fourier Analysis book 2009: Let $(X,\mu)$ be a measure space. Then for all $0 <p,q \leq \infty$, the Lebesgue spaces ...
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2answers
73 views

Does delta distribution remain continuous with respect to quasinorm?

I am thinking the accepted answer which is found here: When viewing $\delta: \mathbb{S} \to \mathbb{R}$ (linear and continuous with respect to the usual semi-norms on the Schwartz-space – or ...
0
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0answers
72 views

Contraction mapping and $L^{2}(\Bbb R \times \Bbb R)$ spaces in inequality

I found this: \begin{equation} \lVert Wx \rVert^{2}_{L^{2}( \mathbb{R} \times \mathbb{R} )} \leq \lVert x \rVert^{2}_{L^{2}(\mathbb{R} )}, \end{equation} which I think ...
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85 views

In the space of distributions, how big is the subspace of functions?

I'm teaching Distribution theory and many of my students still believes that there is only one kind of distribution : the distribution that can be identified to a $L^1_{\text{loc}}$ function. And I ...
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1answer
33 views

Differentiability of smooth functions

The space is from Schwartz space $S$ or the space of all smooth functions of compact support $D$. Can you say anything about how many times the test function is differentiable from this? I think ...
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2answers
136 views

Discontinuity of Dirac Delta distribution

I know that the following holds for Step function, but not sure if it holds for the distribution too. Does the following hold for the Dirac delta distribution too? $\delta$ is a linear functional ...