# 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.

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### Differential operator applied to convolution

Suppose that $g\in \mathcal{S}(\mathbb{R^n})$ (Schwartz space) and $f\in L^p(\mathbb{R^n}).$ The idea is to prove that the differential operator $D^\alpha$ does not follow the Leibniz rule when ...
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### Why convolution regularize functions?

There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
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### Definition of convolution?

Why do we use $x - y$ rather than $x + y$ in the definition of the convolution? Is it just convention? (If we are thinking of convolutions as weighted averages, for instance against "good kernels," it ...
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### Can someone intuitively explain what the convolution integral is?

I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
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### How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
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### How this operation is called?

This operation is similar to discrete convolution and cross-correlation, but has binomial coefficients: $$f(n)\star g(n)=\sum_{k=0}^n \binom{n}{k}f(n-k)g(k)$$ Particularly, $$a^n\star b^n=(a+b)^n$$...
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### Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$A_f(g) = \int f(\tau)g(t-\tau)d\tau$$ and apply it to $g(t)=e^{ikt}$. ...
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### Is $L^2(\mathbb{R})$ with convolution a Banach Algebra?

Is $L^2(\mathbb{R})$ a Banach algebra, with convolution? I am pretty sure the answer is no, because I think that $f,g \in L^2(\mathbb{R})$ does not imply that $f*g \in L^2(\mathbb{R})$. However, I ...
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### $f$ is bounded and continious $\Rightarrow$ the convolution integral $\int f(\tau)g(x-\tau)\text{ d}\tau$ is bounded and continuous

Let $g\in L^1(\mathbb{R}^n)$ and $f:\mathbb{R}^n\to\mathbb{R}$ be bounded and continuous. Why is the convolution integral $$f*g:\mathbb{R}^n\to\mathbb{R}\;,\;\;\;\int f(\tau)g(x-\tau)\text{ d}\tau$$ ...
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### Convolution of functions with compact support

I have a question regarding convolution with compact support: Suppose $f \in L^1(\mathbb{R})$ and $g \in L^p(\mathbb{R})$, and both of them have compact support. Show that $f*g$ (convolution ...
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### convolution of characteristic functions

Suppose $A$ and $B$ are measurable subsets of $\mathbb{R}$ of finite positive measure. Show that the convolution $\chi_A*\chi_B$ is continuous and not identically $0$. Use this to prove that $A+B$ ...
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### Proving $\mu\ast K_n\to\mu$

Let $\{K_n\}$ be approximating unit and $\mu\in M(\mathbb{T})$. Show that $\mu\ast K_n\to \mu$ weakly means $$\int f(t)d(\mu\ast K_n)(t)\to\int f(t)d\mu(t)$$ Suppose $\mu\in L^1(\mathbb{T})$, I know ...
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Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h \omega}{\... 1answer 87 views ### Is it true that f\in W^{-1,p}(\mathbb{R}^n), then \Gamma\star f\in W^{1,p}(\mathbb{R}^n)? I am trying to understand the following paper. In page 1191, in the beggining of the proof of Theorem 2.9. the authors consider the convolution$$v=\Gamma\star f$$They claim that v\in W^{1.p}(\... 2answers 70 views ### Integrability of Maximal Convolution Operator Let f\in L^{\infty}(\mathbb{R}^{n}) be supported in the unit ball and have mean zero. Let \phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n}) be a Holder continuous function with exponent ... 2answers 73 views ### Convolution Theorem and Marginal Density Intuition. In terms of marginal density, how does one know that summing over the x (or rather along the linear line) values for the joint density of (x,z-x) give us the density function of z? More ... 1answer 32 views ### Translations AND dilations of infinite series Sometimes, when working with infinite series, it's useful to add "dilated" or "translated" versions of the infinite series, term by term, back to the original. There are ways of making this rigorous ... 2answers 67 views ### Sums of independent random variables (more than two) [closed] I read that the convolution of two iid random variables is$$(f * g) (z) = \int f(z-y) g(y) dy$$What is the general formula for more than two RVs? For example, for three RVs. 1answer 276 views ### The issue of treating an inverse Fourier transform in terms of a tempered distribution. Consider the wave equation$$ u_{tt}=\Delta{u} \quad u(x,0)=f(x) \quad u_t(x,0)=g(x) \tag{*} $$A solution to this equation is given by$$ u(.,t)=f*\partial_t\Phi_t+g*\Phi_t \tag{**} $$where \Phi_t... 1answer 540 views ### Using Convolution Theorem to find the Laplace transform In previous questions I have used Laplace transform to find the inverse Laplace transform. I have worked through this work booklet (http://www3.ul.ie/~mlc/support/Loughborough%20website/chap20/20_6.... 1answer 170 views ### Proving an integral identity: \int\nolimits_{-\infty}^\infty x f(x)f(t-x) dx =\frac{t}{2} \int_{-\infty}^\infty f(x)f(t-x) dx  Let f be a nonnegative (probably not needed) function on \mathbb{R} such that for all t, xf(x)f(t-x) and f(x)f(t-x) are both integrable in x. Is it true that$$ \int\nolimits_{-\infty}^\...
Let $f,g \in L^2(\mathbb{R^n})$, $\{ f_n \}, \{ g_m \} \subset C^\infty_0(\mathbb{R}^n)$ (infinitely differentiable functions with compact support) where $f_n \to f$ in $L^2$, and $g_n \to g$ in $L^2$....