Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schartz's sense) or measures.

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Exact deconvolution of two matrices using numerical techniques

Suppose that I am given two $n \times m$ matrices $\bf{A}$ and $\bf{C}$, and let $\bf{B}$ be a matrix that is convolved with $\bf{A}$, such that: $\bf{A} * B = C$ In the above, $*$ is the ...
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24 views

convolution of $\delta(t+4) \ast \delta(t-1)$?

How does one solve convolution of $\delta(t+4) \ast \delta(t-1)$ where $\delta$ is dirac delta function? In ordinary function convolution, tricks are obvious but does dirac delta function share the ...
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1answer
36 views

What is the generalization for a convolution in $\mathbb C$?

Since the integration range of "the" convolution is $\mathbb R$, what is a sensible generalization in complex numbers? Would one still integrate over $\mathbb R$, or some other path, or over the ...
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1answer
116 views

Convolution $f*g$ is continuous

Statement: Let $f,g: \mathbb{R}^d \rightarrow \mathbb{R}$ be Lebesgue measurable functions such that $f\in L^1(\mathbb{R}^d)$ and $g\in L^\infty(\mathbb{R}^d)$. The convolution $f*g:\mathbb{R}^d ...
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108 views

Convolution doubt

Can someone explain why the general formula of the convolution is this one: $$(f*g)(t)=\int_{-\infty}^{\infty}f(t-\tau)g(\tau)d\tau$$ But when both $f(\tau)$ and $g(\tau)$ are equal to zero for ...
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380 views

How can you do convolution graphically?

I don't exactly remember whether I should get the common area under the curves of the functions being convolved or I should multiply them and get the area under the resulting curve.
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1answer
29 views

How to obtain the convolution directly (not graphical) of the two functions $e^{-t}u(t)$ and $e^{-2t}u(t)$?

I'm having trouble solving this convolution integral graphically. I don't understand where I stop sliding my function $h(t-\lambda)$ since $x(t)$ doesn't have a boundary as lambda approaches infinity ...
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54 views

Differential equation with fourier transform and convolution

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b. I tried to solve ...
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36 views

Computing a messy convolution

Consider the functions $$ x(t) = u(t - \frac{1}{2}) - u(t - \frac{3}{2}) $$ and $$ h(t) = tu(t) $$ where $u(t) = 1$ if $t \geq 0$ and $u(t) = 0$ if $t < 0$. I'm trying to compute $$ (x*h)(t) ...
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40 views

Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
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1answer
33 views

Another proof of the iniectivity of a linear operator

Let $g(x)= \chi_{[-\frac{1}{2}, \frac{1}{2}]}(x) $, and $ T \colon L^2(\mathbb{R}) \longrightarrow L^2 (\mathbb{R})$ , $Tf= g \star f$. I was asked to prove that $T$ is injective, and I succedeed ...
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1answer
54 views

Inverse Laplace transform of a given function

1) The Laplace transform of f(t) is $\overline{f}(p)=\frac{1}{p}$ when $f(t)=1$ 2) The Laplace transform of $f(at)$ is $\frac{1}{a}\overline{f}(\frac{p}{a})$ 3) The Laplace transform of the ...
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1answer
202 views

The issue of treating an inverse Fourier transform in terms of a tempered distribution.

Consider the wave equation $$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$ A solution to this equation is given by $$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$ where ...
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77 views

Fourier transform of function

What is Fourier transform of $$f(x)=\frac{e^{-|x|}}{\sqrt{|x|}}?$$ I tried to calculate it using $$F(e^{-|x|})=\sqrt{\frac{\pi}{2}}e^{-|a|}$$ and $$F(\frac{1}{\sqrt{|x|}})=\frac{1}{\sqrt{|a|}}$$ and ...
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120 views

Convolution of finite measures

I am puzzled by the following (maybe very stupid) question I stumble upon in the course of a project: let $p$ be a probability measure on some abelian group $E$ (actually, $E=\mathbb{Z}_n$ with its ...
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1answer
83 views

Any clue how to solve this convolution integral?

With other words: find a (closed) expression for $\;\overline{\mbox{sinc}}(x)$ . $$ \overline{\mbox{sinc}}(x) = \int_{-\infty}^{+\infty} \frac{\sin(\omega\xi)}{\omega\xi} ...
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1answer
335 views

Convolution of distributions.

We are given with distributions $f,g \in D'(\Bbb R)$. If $suppf\subset (-\infty,a)$ and $supp(g)\subset(b,\infty)$ then prove that $f*g$ is well defined distribution. where $a$ and $b$ are real ...
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339 views

Differentiability of convolution

First let me say that I have used the search bar and looked through all the "differentiability of convolution" questions that I saw, but none of them seem to cover this case. (If one of them did and I ...
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1answer
379 views

convolution square root of uniform distribution

I need to find a probability distribution function $f(x)$ such that the convolution $f * f$ is the uniform distribution (between $x=0$ and $x=1$). I would like to generate pairs of numbers with ...
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1answer
248 views

Convolution Laplace transform

Find the inverse Laplace transform of the giveb function by using the convolution theorem. $$F(x) = \frac{s}{(s+1)(s^2+4)}$$ If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - ...
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1answer
172 views

How to compute the convolution of two functions which diverge at infinity?

How to compute the convolution of two functions which diverge at infinity? e.g. $e^{x^2}*e^{x^4}$ We can't directly write as $\int_{-\infty}^\infty e^{t^2}e^{(x-t)^4}~dt$ or $\int_{-\infty}^\infty ...
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3answers
1k views

Approximate a convolution as a sum of separable convolutions

I want to compute the discrete convolution of two 3D arrays: $A(i, j, k) \ast B(i, j, k)$ Is there a general way to decompose the array $A$ into a sum of a small number of separable arrays? That is: ...
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365 views

convolution of signals

I'm finding in trouble trying to resolve this exercise. I have to calculate the convolution of two signals: $$y(t)=e^{-kt}u(t)*\frac{\sin\left(\frac{\pi t}{10}\right)}{(\pi t)} $$ where $u(t)$ is ...
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1answer
17 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
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70 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
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26 views

Reference for “Approximation of identity” of a convolution

I need a textbook reference for the "Approximation of identity" of a convolution: http://en.wikipedia.org/wiki/Mollifier#Properties I would appreciate any pointers. The wikipedia article refers to ...
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40 views

Divisor function convolution

I need some help to prove that $$ (d*d)(p^k) = \frac{(k+3)(k+2)(k+1)}{6} \qquad \forall p \in \mathcal{P},\quad \forall k \in \mathbb{N}, $$ where $d$ is the divisor function and $\mathcal{P}$ the set ...
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1answer
47 views

Bound on the derivative of a cut-off function

Let $\rho$ be a smooth function in $\mathbb R^n$ such that $0 \leq \rho \leq 1$ and $\rho$ is supported in the unit disk and let $\rho_\epsilon(x) = \epsilon^{-n}\rho(\epsilon^{-1}\|x\|)$. If $f$ is ...
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1answer
53 views

Convolution operator positive definite?

Let $\mu$ be a compactly supported Borel probability measure on $\mathbb{R}^n$. Consider the convolution operator $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ defined by $$ Tf = f \ast \mu $$ ...
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1answer
54 views

Prove or disprove: $e^{-nG(x)}$, normalized, is an approximation to the identity for $G(x)$ strictly convex

We are given the sequence of functions $$ \phi_{n} = \frac{e^{-nG(x)}}{\int_{\mathbb{R}}e^{-nG(x)}dx}$$ for a nonnegative, strictly convex function $G$ (that is, $G'' \geq c$ for some $c>0$) that ...
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1answer
59 views

Decay of a Convolution

Let $f, g \in L^1\cap L^\infty(\mathbb{R}^d)$ be probability distributions on $\mathbb{R}^d$, and suppose at large $|x|$, $f$ decays like $|x|^{-\alpha}$ while $g$ decays like $|x|^{-\beta}$, with ...
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1answer
43 views

Is $(g \ast f ) '= g'\ast f$ true?

Take $ f \in L^{1} (\mathbb{R})$, and $ g \in L^{\infty}(\mathbb{R})$, with $g$ almost everywhere differentiable and such that $g' \in L^{\infty}(\mathbb{R})$. Prove or disprove: $(f \ast g) \in ...
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74 views

Proof of convolution inequality

I have to prove that if $f$, $g$ $\in L^1(\mathbb{R^n})$ then $\operatorname{dom}\left(f*g\right)$ is a set of full measure and: $\left\|f*g\right\|_{L^{1}} \le \left\|f\right\|_{L^1} ...
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1answer
225 views

Convolution of indicator functions is continuous

Suppose I have an indicator function on a set of measure $E$, which is a subset of $[0,1]$. Is the function of this indicator convoluted with itself a continuous function? How can I show that it is? ...
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50 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
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115 views

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative.

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative. I believe this is convolution. My attempt at the proof: $(f \ast g)(n) = ...
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1answer
160 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
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117 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
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1answer
676 views

Multiplying polynomial coefficients

Take: $u(x)$ and $v(x)$ to be integer polynomials, and then interpret them as sequences in the obvious way: i.e. you put the $i$th term to be the coefficient of $x^i$. Then you'll find that $u\ast ...
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1answer
54 views

A problem about Convolution

If $\phi\in C_0^{\infty}(\mathbb{R}^N)$ and $\psi\in L_{loc}^1(\mathbb{R}^N)$ is defined by $\psi(x)=|x|^{2-N}$, $N\geq 3$ , does $\phi\star\psi$ is in the Schwartz space? Note: $\star$ stands for ...
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607 views

How to compute Inverse Laplace transform using Convolution

How do you evaluate the inverse transform below using convolution ? $$ \mathcal{ L^{-1} } \left[ {\frac{s}{(s^2 + a^2)^2}} \right] $$ I tried $$\begin{align} \mathcal{ L^{-1} } \left[ ...
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1answer
147 views

What does it mean to convolve a matrix with a kernel?

I have a Matrix, M, of dimensions width x height. The problem is to apply the [-1, 0, 1] filter along the x and y axis (i.e. convolve the image with [-1, 0, 1] kernel along horizontal and vertical ...
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1answer
370 views

What is the Fourier transformation of a uniform B-Spline?

I'm looking for the Fourier transformation of the (constant) uniform B-Spline $$N_0(x) = \begin{cases}1 & 0 \leqslant x < 1 \\ 0 & otherwise \end{cases}$$ If $N_0(x)$ would also attain ...
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166 views

Finding spectrum using the convolution property

Using the convolution property, find the spectrum for $$w(t)= \sin(2\pi f_1 t) \cos(2\pi f_2 t).$$ I'm confused on how to solve this question. Can you give me any aproach?
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650 views

FFT with a real matrix - why storing just half the coefficients?

I know that when I perform a real to complex FFT half the frequency domain data is redundant due to symmetry. This is only the case in one axis of a 2D FFT though. I can think of a 2D FFT as two 1D ...
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724 views

Fourier transform of convolution of sinusoidal signals, or product of distributions (generalized functions)

I will unashamedly say that this was at least spurred by homework. However I have gone far beyond the syllabus of the course and still can't find an authoritative answer. And it seems an interesting ...
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956 views

Is convolution in spherical harmonics equivalent to multiplication in the spatial domain?

Spherical harmonic convolution is defined as: $$ ( k \star f )^l_m = \sqrt{ \frac{ 4 \pi }{2l+1} } h^l_m f^l_m $$ I have a function with RGB values for every $(\theta,\phi)$ in the spatial domain. ...
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2answers
2k views

Convolution with multiple step functions

This is a question from Bertsekas' Data Networks. It is question 2.2 on page 141. It is asking for the convolution of the following 2 functions. Function 1: $ s(t) = 1 $, when $0 \leq t \leq T$. It ...
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1answer
2k views

How to sketch the following discrete time signal?

i need to sketch y[n] where * denotes the convolution operator and delta is the unit impulse. I know how to sketch x[n-1] and delta[n-2] but i have problems with the convolution. In my script i only ...
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18 views

Convolution with heaviside function, argument of the heaviside carry on to the dirac function?

So I have this equation to demonstrate: $$ x(t)*u(t)= \int_{-\infty}^t x(\tau)d\tau $$ , where $u(t)=\int_{-\infty}^t \delta(\tau)d\tau$ I opened the convolution as $ \int_{-\infty}^\infty ...