Questions tagged [convolution]

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

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14 views

Convolution of two Cardioid and vonMises PDFs

for the past few days, I've been "on and off" with Mardia's and Jupp's "Directional Statistics" to learn something new about approximating circular distributions. In particular, I've been looking at ...
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19 views

What does this convolution result means?

I tried to predict a time series (for anecdotal purposes, it is a serie of daily detected coronavirus infected cases). For that purpose, I created a complex number for each day, were the real part is ...
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$f, g$ are probability density functions of an normal distribution N(0,1), prove h is $N(0,\sqrt 2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. When $f$ and $g$ are $exp(\lambda)$ it's solved by $\int_{0}^{...
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What is the relationship between the Mellin transforms of $h(x)$, $f(x)$ and $g(x)$ where $h(y)=\int\limits_0^\infty f(x)\ g(y-(x-1))\,dx$

The Mellin transform is defined in (1) below and the standard and alternate Mellin convolutions are defined in (2) and (3) below respectively. (1) $\quad\mathcal{M}[f(x)](s)=\int\limits_0^\infty f(x)\...
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$f, g$ are probability density functions of an exponential distribution, prove h is $\gamma (\lambda ,2)$

I have alredy proved: $f, g$ two density functions. Prove $h(x)=$$\int_{-\infty}^{\infty} g(x-y)f(y) dy$ define a new density function. Then is asked: $f, g$ are probability density functions of an ...
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Analytic and harmonic functions in the unit disc (Question 4.1.1 of “Complex Polynomials” by Sheil-Small)

The question (not homework) is Let $f\in\mathcal{H}$ and suppose that $f(0) = 0$ and $\left|f(z)\right|\leq 1$ for $z\in\mathbb{U}$. Show that $$\left|f(z)\right|\leq \frac{2}{\pi} arg\left(\frac{1+...
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Finding a density function of the sum of two random using the convolution

The following problem is from the book "Probability and Statistics" which is part of the Schaum's outline series. It can be found on page 71 and is problem number 2.74. It is under the section ...
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FIR filter unit step response

Given a FIR filter with impulse response: $$ h(n) = \begin{cases}1, &0 \leq n < 5\\ -1, &10 \leq n < 15 \\ 0, &\text{otherwise}\end{cases} $$ What would be the right approach to ...
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62 views

What is the probability of $P(X+Y=1)$?

$X$~$Exp(1.4)$ and $Y$~$Exp(2.8)$ are independent. What is the probability of $P(X+Y=1)$. I've tried to use the convolution theorem but I couldn't figure it out. Thank you for your help in advance
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Solutions for $ \int_0^{\infty} \frac{1}{\sqrt{t^3\left(t+\tau\right)^3}}\exp\left[- \frac{2t+\tau}{t\left(t+\tau\right)}\right] \, dt $

I am trying to find an analytical solution to the above integral. The context is as follows: I am interested in obtaining an expression for the autocovariance function of the change in groundwater ...
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30 views

Prove that a given distribution is tempered

I have a distribution $E$ such that $\phi \ast E$ is tempered for all $\phi \in C_c^\infty$. Is it possible to prove that $E$ is tempered? Something along the lines \begin{equation} \lim\limits_{n \...
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26 views

Laplace transform of an unusual convolution

I'm trying to solve a (linear, homogeneous) ODE (and find the function f(t)) which includes the convolution \begin{equation} \begin{aligned} (G*f)(t) := \int_{-\infty}^tG(t - u)f(u)du + \int_{t}^\...
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Does convolution of rectangles with different support converge to a gaussian?

Let $Y$ be the unit step function. If I convolve $Y * Y$ I get a triangle, convolve it again and the word "spline" starts to appear in my mind, and finally this $Y * Y * Y * ... * Y$ converges to a ...
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Convolution of a gaussian noise vector with a sinusoid?

I'm trying to understand what a convolution operation does. Right now, I can see that convolution output is high when the signal contains certain frequencies. For example, $f(x) = sin(wx)$ and if I ...
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40 views

auto-convolution of a function from zero to infinity (either manually or by Maple)

I would appreciate it if someone could help me with the following integral and the convolution of the function $f(x)$ with itself from zero to infinity: $$\int_0^\infty xf(x)dx = ?$$ with $$f(x)=\frac{...
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Solving set of sets of mutually exclusive equations derived from convolution and max-pooling

For a research project, I am trying to reconstruct an image by its convoluted and max-pooled result. The result and the kernel weights of various convolution filters are known. I thought to reduce ...
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Convolution theorem for images and kernels

I'd like to understand the convolution theorem for Fourier transforms applied to images and kernels. The theorem states that $F(f *g) = F(f) \cdot F(g)$, where F is the Fourier transform operator, * ...
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Show that $u(x,t)=\int_{\mathbb{R}^n}{\exp\left(-\frac{|x-y|^2}{2t}\right)f(y)dy}$ is $C^\infty$ and converges locally uniformly

Let $f$ bounded continuous function in $\mathbb{R}^n$ then the function $$u(x,t)=\int_{\mathbb{R}^n}{\exp\left(-\frac{|x-y|^2}{2t}\right)f(y)dy}$$ Is $C^{\infty}-$smooth in $\mathbb{R}^n\times\...
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Analytic Number Theory Exercise - The Stability Theorem

This exercise is in the book by P.T. Bateman and H.G. Diamond which i'm not sure whether i perceived it correctly about the integrator involving in the convolution and integral, I was wondering if $$\...
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17 views

Question about convolution of two functions

Is there any function $f(t)$ such its convolution with functions in the form of $g(t) = A(t) cos(\omega t)$ results in $h(t) = A(t) sin(\omega t)$, that is: $$(f\ *\ g)(t)\ =\ \int_{-\infty}^{+\infty}...
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Is there a transform based on Mellin convolution analogous to the Hilbert transform which is based on Fourier convolution?

Fourier convolution is defined as follows. (1) $\quad\left(f(x)\ *_\mathcal{F}\ g(x)\right)(y)=\int\limits_{-\infty}^\infty f(x)\ g(y-x)\ dx$ The Hilbert transform of $f(x)$ defined in (2) below is ...
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Uncertain Transfer Function

$h[n]$ in the picture is convolved with a series of random number $x[n]$. My question is what $h[n]$ describes? Is it a matrix or something else? $$ h[n] = \frac{1}{\sqrt{N}} \underbrace{[1,1,\dots,1]...
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Convolution with an even function

I apologise upfront for the dumb question, but I just totally got stuck with this. An apparent contradiction when taking the convolution with an even function. For two (nice) functions $f$ and $g$ ...
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Spaces of functions which remain linear combination of themselves after being convolved by gaussian?

Say we have the following set of functions $$\mathcal S = \{f_1,f_2,\cdots\}$$ so that for any $n$ there exist uniquely a set of $ \{ c_k \}$: $$(f_n * g)(t) = \sum_{k\in \mathbb Z^+} c_k f_k(t)$$ ...
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39 views

What is the meaning of this symbol [closed]

Could you please explain or give some resources about the symbol "line" on the head of the function
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14 views

Independent Variables in Convolution Derivation

When deriving the convolution representation of a radioactive dumping problem, a professor appears to use both $u$ and $t$ to represent time, as shown in this image: The final result is shown in this ...
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How to implement zero-padded convolution with FFT multiplication?

It is well known that we can implement circular convolution $$t\to(f*s)(t)$$ of two functions $$t\to f(t)\\t\to s(t)$$by utilizing the convolution theorem of Fourier Transforms: $$(f*s)(t) = \mathcal ...
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26 views

Approximating by convolution a function $f \in L^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$.

I'm trying to prove what follows: if $f \in L^1(\mathbb{R})$ such that $\hat{f} \in L^1(\mathbb{R})$, with $\hat{f}(x) = \int_{\mathbb{R}} f(t) e^{-ixt} dt$, and $\phi_n(x) = \frac{n}{\pi}\frac{1}{1+n^...
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How to efficiently calculate $(\sum {\bf M}^T{\bf DM}) {\bf v}$ for diagonal $\bf D$s and convolutional $\bf M$s?

So I am currently working with some matrix-equation systems. I often come across expressions of the type $$\left( \sum_{\forall i} \alpha_i\cdot{{\bf M}_i}^T{\bf D}_i{\bf M}_i \right) {\bf v} $$ ...
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Is it possible to see $L=\int_{x_{1}}^{x_{2}}(\int_{-f(x)}^{f(x)}g(t)dt)dx$ as the convolution between two function?

I have the following integral $L=\int_{x_{1}}^{x_{2}}(\int_{-f(x)}^{f(x)}g(t)dt)dx$ where $f(x)$ is a convex function defined only in the interval $x_1<x<x_2$ and $g(t)$ is a gaussian with rms ...
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39 views

Convolution of x(t) and x(-t)

Consider the signal $x(t)=e^{-t}u(t)$ where $u(t)=\mathbb{1}(t\geq0)$, i.e. the Heaviside function. Find the signal $y(t)=x(t)*x(-t)$ My attempt: $y(t)=x(t)*x(-t)$ $=e^{-t}u(t)*e^{t}u(-t)$ $=e^{-...
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Upper bound for the convolution of two sequences, one of which is finite?

Given $a_i$, $i = 0, ..., n$, the coefficients of a polynomial and $b_0, ...$ the Maclaurin series coefficients of its reciprocal, I am trying to find an upper bound on $$\sum_{i=j-n}^m a_{j-i}b_i$$ ...
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26 views

Derivative of singular convolution integral

While trying to solve a heat conduction problem, I stumbled upon a kind of equation which, in general form, can be written as: \begin{equation} f(t) = \int_0^t \frac{g(x)}{\sqrt{\tau(t)-\tau(x)}}dx \...
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Find the generating function of $\{u_n\}$ where $u_n$ is the probability that the combination $SF$ occurs at $n-1$ and $n$ trial for the first time.

In a sequence of Bernoulli trials let $u_n$ be the probability that the combination $SF$ (success and failure) occurs for the first time at trials number $n-1$ and $n$. Find the generating function. ...
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55 views

Operator norm of convolution operator in L1

I have a convolution operator $T$ in $L^1(\mathbb R)$ defined as $T:f \rightarrow f*g$ , for some $g \in L^1(R)$. I need to prove that $||T||_{L^1}=||g||_{L^1}$. I have found such problem here: Limit ...
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Norm of convolution operator in L1

Consider the convolution operator on $L^1(\Bbb R)$, $f→f∗g$, where $g$ is some $L1$ function. I need to show that the norm of this operator equals to $||g||_1$. I have seen a question here which ...
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find the transmitted code BCH?

for a $(15,7)$ double error correcting BCH code with $G(x) = x^8+x^7+x^6+x^4+1$ if the recieved vector in $r(x)=x^9+x^5+x+1$ using $GF(2^4)$
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Conditional probability of a convolution (sum) of distribution

Suppose X and Y are mutually independent random variables following both normal distributions. I need to calculate the Prob($X+Y \leq z \cap X \leq x$). Clearly, they are not independent events ...
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126 views
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Can convolution be expressed as a differential equation?

The integral equation for (causal) convolution is given by $$y(t) = \int_{-\infty}^{t} K(t - \tau) x(\tau) d\tau$$ Can one write an equivalent differential equation for general well-behaved kernel $...
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Bank Interest in Terms of Convolution

Example 4 from an old set of MIT notes asks: Bank interest. On a savings account, a bank pays the continuous interest rate $r$, meaning that a sum $A_0$ deposited at time $u = 0$ will by time $...
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43 views

How to find a closed form expression for a multivariate convolution identity?

This is a generalization of question Another bizarre convolutional identity. .Let $\lambda^M \ge 0$, $\Lambda \ge 0$, $q \in (0,1)$ and $\theta =2,3,4,\cdots$. In addition to that let $n \in {\mathbb ...
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58 views

Unable to do question 3 in 7.3 from Folland's Fourier Analysis and its Application

I'm unable to answer this question, where we were given $f(x)$: $$f(x)=\begin{cases} 1, & \text{if }-1<x<1 \\ 0, & \text{otherwise}\end{cases}$$ The questions asks me to compute $f*f$ ...
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operator to differentiate a signal over different length intervals

I don't have a deep math background so forgive any in-congruence, I hope you get the idea. I was wondering if there was an operator that would do that: $$ (f(x+y)-f(x))/y $$ $ y $ $ \in $ $]\partial ...
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Using [−∞,0] instead of [0,∞] limit for a convolution difference of independent exponential variables

Let $X_1∼\exp(λ)$ and $X_2∼\exp(λ)$ be two independent exponentially distributed random variables. Find the pdf of $Y = X_1−X_2$ through convolution. My approach: Integrating the product of their ...
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8 views

convolution between normal bivariate and gaussian

i need to perform the convolution between a normal bivariate and a gaussian. Both are normalized. I expect that the result will be a normal bivariate, right ? Which would be the variances of that new ...
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37 views

Show that a convolution of two functions solves an ODE

Given a function $f\in C_c(\mathbb{R})$, meaning $f$ is continuous with compact support in $\mathbb{R}$, and function $\Phi(x) = \frac12|x|$, show that the convolution $u=f \ast \Phi$ is well defined ...
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1answer
36 views

Convolution inequality $||f*g||_{L^1}\le||f||_{L^2}||g||_{L^2} $ in $L^2$

I am having a hard time proving that if f,g$\in{L^2}$, then the convolution g*f(x)= $\int_{y\in \mathbb R}$g(y)f(x+y)dy is continuous. I am stuck trying to show that $||f*g||_{L^1}$$\le||f||_{L^2}||g|...
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1answer
42 views

Problem on the inequality in $L^p$ spaces

The convolution off and $g$ on $\mathbb{R}^d$ equipped with Lebesgue measure is defined by $$f*g(x) = \int_{\mathbb{R}^d} f(x-y)g(y)dy.$$ Define $||f||_{L^\infty} =$ inf$\{ M : |f(x)|< M$ for $\mu$ ...
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31 views

Another bizarre convolutional identity.

Let $\lambda^M \ge0$ and $\Lambda\ge 0$ and $q \in (0,1)$. Now define another three numbers $(a,b,c)$ by solving the following set of non-linear equations below: \begin{eqnarray} c \cdot (c-2) &=&...
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19 views

Convolution operator on $L_\infty(\ell_q)$

Let $\mathcal{S}(\mathbb{R}^{n})$ be the Schwartz space of rapidly decreasing functions on $\mathbb{R}^{n}$ and let $\varphi \in \mathcal{S}(\mathbb{R}^{n})$. I would like to know whether the ...

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