# 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.

2,142 questions
Filter by
Sorted by
Tagged with
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 ...
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 ...
21 views

16 views

19 views

### 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 ...
30 views

### 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, * ...
15 views

11 views

### 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 $$\left( \sum_{\forall i} \alpha_i\cdot{{\bf M}_i}^T{\bf D}_i{\bf M}_i \right) {\bf v}$$ ...
25 views

### 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 ...
39 views

24 views

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 $f*f$ ...
17 views

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: $$(f(x+y)-f(x))/y$$ $y$ $\in$ $]\partial ... 4answers 54 views ### Using [−∞,0] instead of [0,∞] limit for a convolution difference of independent exponential variables Let$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$through convolution. My approach: Integrating the product of their ... 0answers 8 views ### convolution between normal bivariate and gaussian i need to perform the convolution between a normal bivariate and a gaussian. Both are normalized. I expect that the result will be a normal bivariate, right ? Which would be the variances of that new ... 0answers 37 views ### Show that a convolution of two functions solves an ODE Given a function$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$is well defined ... 1answer 36 views ### Convolution inequality$||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}$$\le||f||_{L^2}||g|... 1answer 42 views ### Problem on the inequality in L^p spaces The convolution off and g on \mathbb{R}^d equipped with Lebesgue measure is defined by$$f*g(x) = \int_{\mathbb{R}^d} f(x-y)g(y)dy. Define $||f||_{L^\infty} =$ inf$\{ M : |f(x)|< M$ for $\mu$ ...
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) &=&...
### 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 ...