# Tagged Questions

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.

28 views

19 views

### Meaning of standard integral convolution

In this paper, in the proof of Lemma $13$, there is this sentence: Now, we find a $1$ Lipschitz $\bar{g} \in C^1(\mathbb{R}^n)$ with $\| f - \bar{g}\|_{|\infty} < \epsilon / 2K$, using the ...
39 views

20 views

### Convergence of convolution with an even summability kernel

Suppose that $f(\theta) : [0, 2\pi] \rightarrow \mathbb{R}$ is a monotone increasing real valued function and $\{k_n\}$ is an even summability kernel. I want to show that $f \star k_n$ converges ...
18 views

### Applying the convolution theorem in the presence of a twiddle factor

The convolution theorem says that a 2-d cyclic convolution like $C = U \ast V$ can be evaluated more quickly than doing the raw sum $C_{i,j} = \sum_{a,b}^n U_{a,b} V_{i-a,j-b}$ for each point (assume ...
51 views

### Convolution of measures, why is the notation like this?

In both my book, and on Wikipedia they define convulution of two measures like this: $(\mu_1*\mu_2)(B)=\int_{\mathbb{R}^d}\mathcal{X}_B(x+y)d\mu_1(x)d\mu_2(y)$ It doesn't seem like a typo, but ...
15 views

19 views

### What is the purpose of the requirement on mollification radius?

On page 66 of "Sobolev Spaces (Adams ed2)" in the proof of Lemma 3.16 (Mollification in $W^{m,p}(\Omega)$), it is mentioned that $\varepsilon < {\rm dist}(\Omega', \partial\Omega)$. However, I ...
26 views

### analyze convolution in spatial domain against multiplication in frequency domain

Lets say I have a image of $NxN$ and a separable filter that I want to apply on it. there are 2 ways to do that: 1. By convolution in spatial domain. 2. By multiplication in frequency domain. I need ...
27 views

### Convolution of delta-ish functions

I would like to compute the convolution of a function with itself, where the function is $f(x) = \frac{\delta(x)}{x}$. When there is a shift in the delta function it is easy to compute, but this one ...
36 views

37 views

18 views

### The validity of mollified method to prove the density of $C_0^\infty(\mathbb{R}^n)$

Let $X$ be a function space completed with converge topology (if possible $X$ is normed) such that $C_0^\infty(\mathbb{R}^n)\hookrightarrow X\hookrightarrow L^1(\mathbb{R}^n)$ is continuous dense ...
238 views

### Convolution with a polynomial is a polynomial. Why?

Let $P:\mathbb{R}\to\mathbb{R}$ such that $\deg P=N$. Let $f$, an integrable-$2\pi$-periodic function. Show that $f\star P$ is also a polynomial. So we can prove it for an arbitrary $x^n$ (Since a ...
Let $f : \mathbb{Z}_N \to F$ be a function with a $F$ a field. We say that $f$ has maximum period if the smallest positive integer $r$ (with $r \mid N$) such that $f(j) = f(j + r)$ for all \$j \in \...