Questions on the (continuous or discrete) convolution of two functions.
7
votes
0answers
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 ...
5
votes
0answers
58 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 ...
3
votes
0answers
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 : ...
3
votes
0answers
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 ...
3
votes
0answers
68 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
$$
f(x)= 1,\;\;\;\;\;\;\;\;\;\;g(x) = ...
3
votes
0answers
235 views
A special case of Young's inequality for convolutions
The problem:
Suppose $f,g\in L^1(\mathbb{R})$. Let $x\in \mathbb{R}$ and $\phi_x(y) = f(y)g(x-y)$. Show that for almost all $x$, $\phi_x$ is integrable. For such $x$ let $\psi(x) = ...
2
votes
0answers
22 views
changing the parameters of a function
Lets say we have
$h[n] = ((1/2)^n )(u[n])$
now if we are ask, find h[k-n], then isn't it we should just swapped every 'n' with 'k-n'. So it turns out
$h[k-n] = ((1/2)^{k-n})(u[k-n])$
But why here ...
2
votes
0answers
74 views
clarification asked for 'difference between convolution and crosscorrelation?'
I don't understand answer formulated in ways like this "Thus, $p\ast q$ is the distribution of $X+Y$.
The cross-correlation $p\circ q$ is the distribution $c=(c_n)_n$ defined by ...
2
votes
0answers
52 views
Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
2
votes
0answers
51 views
Convolution and Smoothness Conditions
Suppose $f(x),g(x)\in L_1(\mathbb{R})$, with both $|f(x)| \leq 1$, $|g(x)| \leq 1$ and $|f(x)| \rightarrow 0$, $|g(x)| \rightarrow 0$ for $|x| \rightarrow \infty$. Given that we have two other ...
2
votes
0answers
45 views
Show compactness of an evolution operator
Consider the heat equation
$$
u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$
with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$
and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.
1.) ...
2
votes
0answers
68 views
When does $|f*g|_{p}=|f|_{1}|g|_{p}$?
From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4
Suppose $1\le p\le \infty$, $f\in L^{1}(\mathbb{R}^{1})$, $g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ...
2
votes
0answers
45 views
bound on Hilbert transform
Consider $\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where $m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e. $T$ is the operation of taking Fourier transform and multiplying with the function $m(\xi)$. I am ...
1
vote
0answers
41 views
Fubini theorum for integrating 1 dimension of a 3d convolution
I have 3D volume that is convoluted with a 3D blur function. Both are positive and integrate to a finite value. I can see experimentally (meaning playing with matlab) that this is true:
$\int_{-a}^{a ...
1
vote
0answers
28 views
Cancellation of summations
I am working on some stuff related to the convolution property of the discrete Fourier transform. If we consider:
$$\sum_{p = 0}^{N-1}\hat{s}_{p}e^{ik_{p}x_{m}} = \sum_{p = ...
1
vote
0answers
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 $f$, ...
1
vote
0answers
48 views
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 ...
1
vote
0answers
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
$$
w(n) ...
1
vote
0answers
72 views
Poisson exponentiation distribution family and convolution
Assume $\xi_i \sim \mathbb{F}_{\lambda_i}(x)$ are random variables from Poisson distribution.
Consider random variables $\eta_i \sim \tilde{F}_{\lambda_i,t}(x)$, where $\tilde{F}_{\lambda_i,t}(x) = ...
1
vote
0answers
70 views
Convolution with a special approximation to the identity function
I'm working my way through Stein and Shakarchi's Real Analysis, and I'm having some trouble figuring this exercise out.
Given the function $K_\delta$ that satisfies the normal approximation to the ...
1
vote
0answers
65 views
Consider the correlation of two functions, what is the derivative of the result with respect to one of those functions?
I have a problem that comes up from time to time in signal processing applications.
Let $f(x)\geq0\, \forall x$ and $g(x)$ be real functions with finite range and support.
Let $I(f(x),g(x)) = ...
1
vote
0answers
54 views
Integrability and differentiability of convolution of the fundamental solution and an integrable function
Define a function $\Gamma(\cdot)$ as
$$
\Gamma(x-y)=\frac{1}{2\pi}\log\|x-y\|,\quad x\neq y
$$
where $x=(x_1,x_2),y=(y_1,y_2)\in R^2$, and $\|x-y\|^2=(x_1-y_1)^2+(x_2-y_2)^2$. Note that $\Gamma(x-y)$ ...
1
vote
0answers
113 views
Cross Correlation
The cross-correlation function is defined as follows if $\bar{f}$ is the complex conjugate of $f$ and we assume that $f$ is real, such that $\bar{f} = f$.
$$
\begin{align}
f \star g &= ...
1
vote
0answers
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) , ...
1
vote
0answers
176 views
Convolution of two functions
for example I have somethink like that,
$$
\begin{align*}
f(x) &=
\begin{cases}
\frac{1}{3}x - \frac{2}{3} &\text{where }2 < x \leq 4, \\
\frac{-2}{3}x + \frac{10}{3} ...
1
vote
0answers
68 views
Solution for this Convolution
We have $f(z)=z+ \sum_{n=2}^{\infty} a_{n}z^{n}$ where $a_{n}$ is a constant and $g(z)=z$, $(f*g)(z)$ is equal to what? i still wondering to confirm that $(f*g)(z)=z$.
1
vote
0answers
190 views
FFT signal post processing
This is more a "post a suggestion" topic rather than a question. And thank you if you are willing to read this whole.
I've been studing the code in the Nvidia Cuda SDK regarding how to operate a ...
0
votes
0answers
19 views
Support of the convolution of two test funtions.
If $g\in C^{\infty}_c$ defined on $\Bbb R^n$ and K is the support of function $g$. I want to find the support of $g_\epsilon$. Where $g_\epsilon$ is regularization of $g$.
Regularization of $g$ is ...
0
votes
0answers
6 views
Closeness of a family of function under convolution.
I'm interested in functions defined over the non-negative integers that are a product of an exponential function and a polynomial. So a standard term of such a function is something like
$$
f(k) = ...
0
votes
0answers
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(?)
...
0
votes
0answers
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 ...
0
votes
0answers
24 views
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 ...
0
votes
0answers
34 views
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 ...
0
votes
0answers
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):
...
0
votes
0answers
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 ...
0
votes
0answers
130 views
Convolution of two functions (pdfs)
I want to convolve two signals . The range of each of the signal is 0 to 1.
...
0
votes
0answers
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 ...
0
votes
0answers
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?
0
votes
0answers
37 views
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 ...
0
votes
0answers
39 views
Extension of Convolution theorem
Is it possible to extend the convolution theorem to convolve tensors, as we do with discrete matrices?
0
votes
0answers
54 views
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.
...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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:
...
0
votes
0answers
26 views
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 $BASE-1, \quad BASE \in \mathbb{n},\quad BASE > 1$.
...
0
votes
0answers
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 ...
-1
votes
0answers
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 ...
