# Tagged Questions

Questions on the (continuous or discrete) convolution of two functions.

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183 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 ...
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### 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 ...
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87 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 : ...
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74 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 ...
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### 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) = ...
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### 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) = ...
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22 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 ...
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### clarification asked for 'difference between convolution and crosscorrelation?'

I don't understand answer formulated in ways like this "Thus, $p\ast q$ is the distribution of $X+Y$. The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by ...
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52 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 ...
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51 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 ...
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### 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.) ...
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### 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 ...
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45 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 ...
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41 views

### Fubini theorum for integrating 1 dimension of a 3d convolution

I have 3D volume that is convoluted with a 3D blur function. Both are positive and integrate to a finite value. I can see experimentally (meaning playing with matlab) that this is true: \$\int_{-a}^{a ...
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28 views

### Cancellation of summations

I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider: \sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
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81 views

### convolution of L1 function with a harmonic oscillation

I have to show that the convolution of a function $f \in L^1(\mathbf{R})$ with the harmonic oscillation $\phi_\omega (t) = \exp(2 \pi i t \omega)$ is equal to the Fourier Transform of $f$, ...
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### Convolutions of Path Integrals of Gaussian Functions

I was looking at a question on a physics forum (http://physics.stackexchange.com/questions/45955/splitting-light-into-colors-mathematical-expression-fourier-transforms) and I wanted a more ...
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41 views

### Maxium value of discrete convolution

I'm trying to calculate the maximum possible short-term energy $E[n]$ of a sampled signal $s$ in terms of $N$ and $\text{bitdepth}$. $$E[n] =\sum_{m=-\infty}^{\infty} s^2[n]w[n-m]$$ where  w(n) ...
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72 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) = ...
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70 views

### Convolution with a special approximation to the identity function

I'm working my way through Stein and Shakarchi's Real Analysis, and I'm having some trouble figuring this exercise out. Given the function $K_\delta$ that satisfies the normal approximation to the ...
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### Consider the correlation of two functions, what is the derivative of the result with respect to one of those functions?

I have a problem that comes up from time to time in signal processing applications. Let $f(x)\geq0\, \forall x$ and $g(x)$ be real functions with finite range and support. Let \$I(f(x),g(x)) = ...
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### Integrability and differentiability of convolution of the fundamental solution and an integrable function

Define a function $\Gamma(\cdot)$ as $$\Gamma(x-y)=\frac{1}{2\pi}\log\|x-y\|,\quad x\neq y$$ where $x=(x_1,x_2),y=(y_1,y_2)\in R^2$, and $\|x-y\|^2=(x_1-y_1)^2+(x_2-y_2)^2$. Note that $\Gamma(x-y)$ ...
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### Cross Correlation

The cross-correlation function is defined as follows if $\bar{f}$ is the complex conjugate of $f$ and we assume that $f$ is real, such that $\bar{f} = f$.  \begin{align} f \star g &= ...
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102 views

### n-th self discrete convolution

Lets define discrete $f_N(i) = 1,\space i = 1...N$ I need to find $G_N^m = \underbrace {f_N * f_N * ... * f_N}_{m}$ For example $G_6^3$ have value (1,3,6,10,15,21,25,27,27,25,21,15,10,6,3,1) , ...
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176 views

### Convolution of two functions

for example I have somethink like that,  \begin{align*} f(x) &= \begin{cases} \frac{1}{3}x - \frac{2}{3} &\text{where }2 < x \leq 4, \\ \frac{-2}{3}x + \frac{10}{3} ...
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68 views

### Solution for this Convolution

We have $f(z)=z+ \sum_{n=2}^{\infty} a_{n}z^{n}$ where $a_{n}$ is a constant and $g(z)=z$, $(f*g)(z)$ is equal to what? i still wondering to confirm that $(f*g)(z)=z$.
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190 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 ...
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19 views

### Support of the convolution of two test funtions.

If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$. Regularization of $g$ is ...
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### Closeness of a family of function under convolution.

I'm interested in functions defined over the non-negative integers that are a product of an exponential function and a polynomial. So a standard term of such a function is something like  f(k) = ...
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21 views

### Convolution of logistic function and gaussian distribution

I am trying to solve the folowing problem: $$\int \exp\left(-\frac{(x-u)^2}{2\sigma^2}\right) \log(1+\exp(ax + b)) \,dx$$ which I think is very complicated and there is no closed form solution(?) ...
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21 views

### Convolution of Inverse Gaussian distribution

Im having problems with showing that the sum of two inverse gaussian distributed random variables are stable under convolution, i.e. let \$f_{T_a}(t) = \frac{a}{\sqrt{2 \pi t^3}} e^{-\frac{(a- \nu ...
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### Inclusion regarding the support of the convolution of two functions

Let $u \in L^1(\mathbb R^n)$ and $v \in L^p(\mathbb R^n)$ where $1 \le p \le \infty$. Show that $$\textrm{spt}(u \ast v) \subseteq \overline{\mathrm{spt}(u) + \mathrm{spt}(v)}$$ where the addition ...
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### Discrete convolution: where do I go from there?

I took this from the book Signals and Systems by Haykin. I have the following discrete system: $y[n] = u[n]*u[n-3]$, where $*$ is the discrete convolution and $u[n]$ is the unit step function, and ...
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31 views

### Strange convolution equation

In an article ( https://www.dropbox.com/s/3012v4s1ngpimvg/gridding_Schomberg_Trimmer.pdf ) about implementation of Gridding method for parallel-beam tomography there's an equation(#47 in the article): ...
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45 views

### Obtaining Impulse Response from Graph

I want to know how to solve those types of problems.. is it by inspection ? Consider the linear system below. When the inputs to the system $x_1[n]$, $x_2[n]$ and $x_3[n]$, the responses of the ...
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130 views

### Convolution of two functions (pdfs)

I want to convolve two signals . The range of each of the signal is 0 to 1. ...
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27 views

### Distribution function approximation: Poisson exponentiation

I want to find normal approximation of Poisson exponentiation distribution. Okay, some introduction to problem: Assume that $\xi_i \sim F_{\lambda_i}(x)$ - Poisson distribution' random variables ...
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76 views

### What is the purpose and usage of convolution?

I am curious of what the purpose and usage of convolution are. Why is convolution created? In layman's term (and in mathematical term), what defines convolution?
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### Vestigial Filter, find modulated signal?

I have been stuck on this question for a while now. It has to do with vestigial sideband. I wasn't sure if I should be dividing $H(\omega)$ graph values by 2 because only the positive side of the ...
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39 views

### Extension of Convolution theorem

Is it possible to extend the convolution theorem to convolve tensors, as we do with discrete matrices?
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### Weighted convolution?

Suppose I have two arbitrary discrete probability distributions with the same domain. I want to convolve the two together to come up with third distribution, however I want them to be weighted. ...
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327 views

### how does one convolve two matrices

so in OpenCV I retrieve a Gabor kernel for image processing which is a 10:10 matrix. I have a gray matrix of the original image. How do I convolve the two and get the output of the convolution? I'm ...
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83 views

### Convolutions, Compact Support and the Divergence Theorem

Ok, first off, this is a long question, so apologies for that. My LateX isn't up to par, so I've coded what I can and I've linked the rest. I've just proved this, and it probably leads on to the bit ...
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63 views

### convolution related question: shifting?

i was wondering, for convolution, when we do the graph shifting for h(t-tou) we flip the graph on the y axis and then if t = 0.5, then shouldn't we shift the graph left by 1/2? In the examples I am ...
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198 views

### Numerical solution of an integro-differential equation with convolution

I have an integro-differential equation that I need to solve numerically. The equation is of the form: $$\frac{dX}{dt} = cX - X\left(b + q\frac{dX}{dt}\right),$$ where $q\frac{dX}{dt}$ denotes the ...
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238 views

### Circular to linear convolution with matrices

I know how to perform a circular convolution with vectors (http://engineering-matlab.blogspot.it/2010/12/matlab-program-for-implementing_5864.html) and I know that circular convolution can be obtained ...
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187 views

### Convolution between a kernel and an image with FFT

In the FFT2D paper (Fast Fourier transform used for a convolution with a kernel in the frequency domain), I'm lost at the second page first picture: ...
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### An argument for error accumulation during complex DFT

I am doing FFT-based multiplication of polynomials with integer coefficients (long integers, in fact). The coefficients have a maximum value of $BASE-1, \quad BASE \in \mathbb{n},\quad BASE > 1$. ...
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169 views

### Convolution problem

Hi i am really stuck trying to do this convolution in order to find zero state response. The convolution table only contains $(e^t)u(t)$ not $u(-t)$ can someone show me the steps with some brief ...
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14 views

### Uniform Convergence of Convolutions

$\displaystyle Q_n(t) = ne^{-nt}$ $Qn(t)*f(x) = \int_0^{\infty} Q_n(t)*f(x-t) dt$ How do I prove that this converges uniformly? It seems to be similar to the proof of the Weierstrass ...
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