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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8
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0answers
215 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
7
votes
0answers
84 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
4
votes
0answers
50 views

Finding convolution identites

Suppose I have the following definition: $$\frac{x^2/2!}{e^x-1-x}=\sum_{k=0}^{\infty}A_k\frac{x^k}{k!}$$ I want to find a convolution identity for these coefficients $A_k$, but I've never studied ...
4
votes
0answers
98 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
4
votes
0answers
100 views

Are any of those quotient rings isomorphic to other well known rings?

(1) Let $C_b(\mathbb{R})$ be the ring (with pointwise multiplication and addition) of bounded continuous functions. Let $I_0=\{f_{(x)} \in C_b(\mathbb{R}) \space | \space lim_{x \to \pm ...
4
votes
0answers
81 views

Need a fast algorithm of adaptive convolution

Good morrow, gentlemen! I have to apply some kind of adaptive filter to my function $f(x).$ I present each point of my signal as a Gaussian, whose bandwidth depends on its location (not the point of ...
4
votes
0answers
329 views

Why matrix representation of convolution cannot explain the convolution theorem?

A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
4
votes
0answers
539 views

Infinite self-convolution for a function

I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function $f : ...
4
votes
0answers
124 views

When does $|f*g|_{p}=|f|_{1}|g|_{p}$?

From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
3
votes
0answers
56 views

Convolution-like operator on (probability) measures on $[0,1]$ yielding measures on $[0,1]$.

Is there a "correct" or "best" way to define convolution of two (Borel) probability measures on $[0,1]$ to yield another probability measure on $[0,1]$? Recall that the convolution, $\mu * \nu$, of ...
3
votes
0answers
116 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
3
votes
0answers
47 views

changing the parameters of a function

Lets say we have $h[n] = ((1/2)^n )(u[n])$ now if we are ask, find h[k-n], then isn't it we should just swapped every 'n' with 'k-n'. So it turns out $h[k-n] = ((1/2)^{k-n})(u[k-n])$ But why here ...
3
votes
0answers
178 views

The norm of an operator

Let $\rho(x)$ be a weight function in a unit sphere, such that \begin{equation} \begin{array}{l} \displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\ \displaystyle 2. \rho(x)\in ...
3
votes
0answers
254 views

Convolution Exercise Homework

Put $\varphi(t)= 1- \cos \;t\;\;\;$ if $\;\;\;0 \leq t \leq 2 \pi$, $\varphi(t) = 0$ for all other real $t$. For $-\infty < x < \infty $, define $$ f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
3
votes
0answers
286 views

Convolution theorem in 3D

Suppose to have a 3-dimensional discrete grid. I would like to convolve it with a 3-dimensional tensor (a 3x3x3 "cube"), applying the convolution theorem. Hence, I should apply a Fourier transform to ...
3
votes
0answers
441 views

A special case of Young's inequality for convolutions

The problem: Suppose $f,g\in L^1(\mathbb{R})$. Let $x\in \mathbb{R}$ and $\phi_x(y) = f(y)g(x-y)$. Show that for almost all $x$, $\phi_x$ is integrable. For such $x$ let $\psi(x) = ...
2
votes
0answers
20 views

Help proving this Convolution Integral:

Can someone give me the steps for proving the following integral: ...
2
votes
0answers
90 views

Convolution of two Gaussians or two sinc functions using direct integration

I tried to solve the following to problems from Gaskil's book Linear Systems, Fourier Transforms, and Optics. But I'm struggling to get the right results. My experience with calculating convolutions ...
2
votes
0answers
36 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
2
votes
0answers
48 views

Do asymptotically equivalent coefficients survive convolution at least in Θ?

This is a follow-up question to this one where I asked if asymptotic equivalence of coefficients carried over after convolution, resp. why this was not the case. Answerer Daniel Fischer proposed that ...
2
votes
0answers
56 views

Expectation and convolution question.

I am learning in an image processing course, and the professor did the following: As part of a derivation, has this: What I do not understand, is how he was able to remove $r(i,j)$ to the ...
2
votes
0answers
54 views

Is this function monotonically non-decreasing?

I am wondering if the function $L[n]$ defined on $n=0,1,2,\ldots,N$ below is "monotonically" non-decreasing in $n$. I put monotonically in quotes because the function is not continuous and I am not ...
2
votes
0answers
165 views

Need help with the convolution of two complex functions

Could someone start me off with how to find the convolution of these two functions? Using the normal equation for convolution seems impossible as a common overlap interval is required for ...
2
votes
0answers
130 views

Show that convolution satisfies partial differential equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
2
votes
0answers
48 views

Inverse Fourier transform to get convolution

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$ Let ...
2
votes
0answers
418 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
2
votes
0answers
50 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...
2
votes
0answers
220 views

Proof of a method to find the points of maximum slope

According to method described in a paper [1] if we want to find points of maximum slope in a signal $f(t)$, then one has to do following Convolve $f(t)$ with $g(t)$ where $g(t)=-cos(\omega ...
2
votes
0answers
251 views

Convolution of compactly supported functions

Let $f,g : \mathbb R \rightarrow \mathbb R$ continuous and compactly supported. I want to show that $f*g$ is continuous and compactly supported. I am 100% sure how to do it. I began as follows: ...
2
votes
0answers
80 views

Solving Dirichlet problem by means of potential theory

Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and consider the Dirichlet problem with $f\in H^{-1}(\Omega)$ $$\tag{1}-\Delta u=f$$ Is there a way to solve this problem by using ...
2
votes
0answers
300 views

How to express multiplication of two spherical harmonics expansions in terms of their coefficients?

Consider a spherical harmonics expansion/series like this: $$f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)$$ Presumably if we take two functions on ...
2
votes
0answers
106 views

General approach to prove the smoothness of convolution

Consider $\mathbb{R}^d$ and $D_i = \partial /\partial x_i$, in many cases $$D_i(f*g) = D_if*g,$$ given one of $f,g$ is smooth and the other is $L^p$ integrable. I am wondering if there is a general ...
2
votes
0answers
91 views

Convolution and Smoothness Conditions

Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other ...
2
votes
0answers
65 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
2
votes
0answers
216 views

Poisson exponentiation distribution family and convolution

Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution. Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
2
votes
0answers
80 views

bound on Hilbert transform

Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
2
votes
0answers
249 views

FFT signal post processing

This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole. I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
1
vote
0answers
25 views

Equality condition for convolution's $L^p$ norm.

Suppose that $1< p< \infty$, $f\in L^1(R)$, and $g\in L^p(R)$ and that $\|f*g\|_p=\|f\|_1\|g\|_p$. Show that then either $f=0$ a.e or $g=0$ a.e I have solved for $g=0$ a.e. if $||f||_1>0$ ...
1
vote
0answers
42 views

convolve chirp with rect

I'm trying to evaluate $$g[x] = f[x] \ast f[x]$$ where * is the convolution operator and $$f[x] = RECT(\frac{x-2.5}{5}) \cdot exp (+i \pi x^2)$$ I assume the best approach to this equation is: ...
1
vote
0answers
37 views

Maximum of a convolution

I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function ...
1
vote
0answers
11 views

Giving a bound for |f(x) \star \phi_k(x) -f(x)|

Here is the problem: Let $\phi(x) \in S$, where $S$ is the Schwartz class, such that $\displaystyle\dfrac{1}{\sqrt{2\pi}}\int_{\mathbb{R}} \phi=1$. Also, for some $N\in\mathbb{N}$, ...
1
vote
0answers
25 views

2D convolution: how to eyeball it?

I have a question of doing simple convolution in 2d by just "eye-balling" it without doing the actual computation. In 1D discrete time, when we have a simple input ...
1
vote
0answers
38 views

Sort of Convolution

I was wondering if the following convolution I am considering already has a name or well-studied. Thanks for your help. If the condition of the positive radius of convergence is not enough, then I ...
1
vote
0answers
35 views

Evaluating convoluted integrals of complex exponentional and rational

I want to evaluate the following integral: \begin{equation} f_{abcd}(t) = \int_{-\infty}^{\infty}d\lambda\int_0^{t-\lambda} d\tau \frac{e^{i a \tau}}{ (b+i \tau)^{5/2} } \int_0^{t-\lambda} d\tau ...
1
vote
0answers
26 views

L^2 space convolution inequality

How do you use Young's Inequality, and all these convolution formulae to prove the following inequality $$||f*g||^2_{L^2(\mathbb{T})}\leq ||f*f||_{L^2(\mathbb{T})}||g*g||_{L^2(\mathbb{T})}$$ where ...
1
vote
0answers
67 views

Distribution of the sum of many lognormal random numbers from same distribution

In my application I have to sum up a lot (between 1000 and 2000) lognormally distributed random numbers and use their sum. All random numbers that I sum up follow the same distribution. The current ...
1
vote
0answers
34 views

understanding discrete-time convolution

I'm trying to understand the discrete-time convolution for LTIs and its graphical representation. standard explanations (like: this one) start with the idea of decomposing an input signal $x[t]$ into ...
1
vote
0answers
189 views

Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$ f*f=g\cdot f. $$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
1
vote
0answers
61 views

Absolute continuity and convolution

Suppose that $\mu$ is a finite Borel measure on the real line, $f, g\in L^1(\mu)$. Define $\nu=\mu\ast\mu$. Do I understand correctly that the convolution $f\mu\ast g\mu$ is absolutely continuous wrt ...
1
vote
0answers
50 views

What is $D\delta$ if $D$ is ordinary differential operator and $\delta$ is the Dirac distribution?

I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of ...