Articles of convolution

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 $\alpha>0$. Define an approximation to the identity $\left\{\phi_{t}\right\}_{t>0}$ by $\phi_{t}:=t^{-n}\phi(\cdot/t)$, and consider the maximal convolution operator $M_{\phi}$ defined by $$M_{\phi}(f)(x)=\sup_{t>0}\left|(f\ast\phi_{t})(x)\right|, \qquad\forall x\in\mathbb{R}^{n} \tag{1}$$ If $\phi$ is compactly supported in a ball […]

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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 importantly, can someone explain this intuitively? (aside from proofs)

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 for each. If you want to add a “translated” copy of a series to the original one, you can: attach the series coefficients […]

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 $v\in W^{1.p}(\mathbb{R}^n)$. Does anyone knows why this is true with $f$ being only in $W^{-1,p}(\mathbb{R}^n)$? Update: Maybe this can thrown some light upon the matter. […]

Relation between the two probability densities

Suppose $X_1, X_2, Y_1, Y_2$ are independent random variables on the same probability space with densities $f_1,f_2,g_1,g_2$ respectively. If $$ \int_{x} f_1(x)f_2(z-x) \,dx = \int_{y} g_1(y)g_2(z-y) \,dy \quad (*)$$ for all feasible $z$ and $$ \frac{f_i(x)}{g_i(x)} $$ is non-decreasing in $x$ for all $x$ in the support of $X_i$ and $Y_i$ for both $i\in\{1,2\}$ then […]

Convolution that is equiprobable to a uniform distribution

I am struggling with finding two distributions a and b on the non non-negative integers (both not concentrated at 0, this is t trivial) such that the convolution of a and b is the equiprobable distribution on the set 0,1,…n-1 n is not a prime number. Any ideas?

Convolution of half-circle with inverse

I am trying to compute the function: $$f(\lambda)\equiv\int_{-1}^{1}\frac{\sqrt{1-x^2}}{\lambda-x}dx.$$ It arises as the convolution of the semi-circle density with the inverse function. When $\lambda\in(-1,1)$ it can only be defined as a Cauchy Principal Value. I have a hunch that I need to go into the complex plane to solve this, but am not sure how to […]

Multiple self-convolution of rectangular function – integral evaluation

I am trying to find an $n$-multiple convolution of a rectangular function with itself. I have a function $f(x) = 1$ for $0<x<1$, 0 otherwise. I define $$ g_2 (y) = \int_{-\infty}^{\infty} f(y+x) f(x) \mathrm{d} x $$ and recursively $$ g_n (y) = \int_{-\infty}^{\infty} g_{n-1}(y+x) f(x) \mathrm{d} x \, . $$ To make it easier […]

Convolution of an integrable function of compact support with a bump function.

Let $f\in L^1(\mathbb{R})$ be of compact support and $\psi(x)=C \exp(-(1-x^2)^{-1})$ where $C$ is chosen so that $\int_{\mathbb{R}} \psi =1$. Show that the convolution $f*\psi(x)=\int_{\mathbb{R}} f(x-y)\psi(y) dy$ is infinitely differentiable.