# Tagged Questions

42 views

### problem with convolution

I'm struggling with this kind of problem: I have an assumption that $f$ and $g$ are in $L^2(R)$, and I should prove that $f\star g \rightarrow 0$ when $|x| \rightarrow \infty$. I think (but I'm not ...
24 views

### When convolution of two functions has compact support?

It is well-known that, if $f$ and $g$ are compactly supported continuous functions, then their convolution exists, and is also compactly supported and continuous (HÃ¶rmander 1983, Chapter 1). Next, ...
55 views

### Convolution with itself equals itself times a function

Consider the case that $f \in L^1(\mathbb{R})$ and $g \in L^1_{loc}(\mathbb{R})$. Then look at the equation $$f*f=g\cdot f.$$ I know that if $g$ is constant, then $f=0$. But what about other $g$'s? ...
28 views

### Convolution of Strictly Convex Function

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be a $C^2$, strictly convex function, and $\theta_\epsilon$ the standard approximation to identity ...
42 views

### convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
19 views

### Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
13 views

### convolution of three functions of two variables

Give three functions of two variables $a(x,y),b(x,y),c(x,y)$ one can construct the following convolution like integral: $y(x,y) = \int dx' dy' a(x',y')b(x-x',y') c(x-x',y-y')$ which I have a hard ...
37 views

### estimating $L^p$ norm of $\frac{x}{|x|^3} \ast (-\text{div}_x \ldots)$

I've been having problems with following an argument in a paper I'm reading, I hope someone can help me understand the point. Suppose $f(t,\cdot,\cdot)$ is a $C_0^1 (\mathbb{R}^6)$ function for all ...
39 views

### Bound on uniform norm of convolution of $L^p$ functions

This is Proposition 8.8 in Folland's Real Analysis: If $p$ and $q$ are conjugate exponents, $f \in L^P$, and $g \in L^q$, then $f*g(x)$ exists for every $x$, $f*g$ is bounded and uniformly ...
25 views

### Min $+$ convolution is associative

Although the following question was encountered in a Communication Networking textbook, the problem is still one of algebraic and analytic manipulation. Define the (min,+) convolution of two real ...
71 views

### Mollification of a function continuous on $\mathbb{R}\backslash\{0\}$, need uniform convergence

We know that if $f:\mathbb{R} \to \mathbb{R}$ is continuous, then its mollification $f_\epsilon$ converges uniformly to $f$ on compact subsets of $\mathbb{R}$ as $\epsilon \to 0$. My question is, ...
25 views

145 views

### Convolution of indicator functions is continuous

Suppose I have an indicator function on a set of measure $E$, which is a subset of $[0,1]$. Is the function of this indicator convoluted with itself a continuous function? How can I show that it is? ...
65 views

Let $f$ be a bounded measurable function with support on the unit disk $\mathbb D \subset \mathbb R^2$ and let $g$ be an analytic function on $\mathbb R^2$. Is it true that the convolution $h = f ... 2answers 136 views ### Let$S$be the Schwartz class. Show that if$f,g\in S$, then$fg\in S$and$f*g\in S$, where$*$denotes convolution. Let$S$be the Schwartz class. Show that if$f,g\in S$, then$fg\in S$and$f*g\in S$, where$*$denotes convolution. To differentiate$fg$, we may apply Leibniz's rule ( ... 0answers 71 views ### Is the Convolution of a Schwartz Function with an$ L^{p} $-Function a Smooth$ L^{p} $-Function? Let$ n \in \mathbb{N} $and$ p \in \mathbb{R}_{\geq 1} $. If$ f \in \mathscr{S}(\mathbb{R}^{n}) $and$ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the ... 0answers 32 views ### Convolution of complex-valued probability distributions This may be an elementary question, but I am wondering: suppose that I have two complex-valued random variables$X$and$Y$with corresponding density functions$f_X(x)$and$f_Y(y)$. Obviously ... 1answer 134 views ### Convolution and uniform continuity If$f\in L^{\infty}(\mathbb{R}^n)$and$f$is continuous at$x$, then $$\lim_{k\to\infty}(f*\phi_k)(x)=cf(x)$$ If$f\in L^{\infty}(\mathbb{R}^n)$and is uniformly continuous, then$f*\phi_k\to cf$... 1answer 74 views ### Interesting inequality$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$over$L^p$Consider the function $$F(x)=\int_0^\infty \frac{f(y)}{x+y} \, dy, \quad0<x<\infty$$ Prove that if$1<p<\infty$, $$\|F\|_p\le \frac{\pi}{\sin(\pi/p)}\|f\|_p$$ and show that the constant ... 1answer 61 views ### Give an example of$f\in L^1$,$g\in L^{\infty}$, such that$f*g\notin C_0$(meaning$\lim_{|x|\to\infty}(f*g)(x)\neq0$) Give an example of$f\in L^1$,$g\in L^{\infty}$, such that$f*g\notin C_0$(meaning$\lim_{|x|\to\infty}(f*g)(x)\neq0$) Here's a theorem from my real analysis book: Assume$1\le p\le \infty$and ... 1answer 129 views ###$L^pSpaces, Young's Theorem, Convolutions, and Minkowski's Inequality I need to show \begin{align} \|f*g\|_p \le \|f\|_p\|g\|_1 \end{align} By using the generalized Minkowski inequality instead of just Young's Theorem. I have spent a lot of time, but I keep hitting a ... 1answer 33 views ### Convolution composed with an invertible matrix LetT$be an invertible$n \times n$matrix and let$(h \circ T)(x)$mean$h(Tx)$. Take functions$f,g$. Does it hold that$(f*g) \circ T = |det(T)| (f \circ T) * (g\circ T)?$I have had some ... 1answer 69 views ### Does negative distributive property of convolution over cross correlation holds? Let$\star$denote convolution binary operation and$\otimes$denote cross correlation binary operation between two functions. Let$f,g,h$be functions. Does this negative distribution property ... 1answer 116 views ### Convolution of an$L^1$function and a function that tends to$0$results in a function that tends to$0$I'm trying to solve the following problem in review for a test, but have only partly succeeded: Let$K \in L^1(\mathbb{R})$and$f$be a bounded, measurable function on$\mathbb{R}$, with ... 0answers 47 views ### Weakest Conditions for Convolution to be Differentiable I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose$f \in C^{1}(\mathbb{R})$. Then what conditions on the function$g$would ensure that ... 1answer 57 views ### Asymptotics at the origin of the convolution with an approximation to the identity. In short, I am trying to find sufficient conditions for an approximation to the identity function$K_h$so that, for$h$small enough and fixed, the asymptotics at the origin of an$L^1 \cap L^2$... 0answers 184 views ### What's the exact definition for convolution? I tried to solve the problem in Stein's Real analysis, 1ed, P94, Ex 21 (c), which asked to show that for any two measurable functions$f,g$on$R^d$, the convolution of$f$and$g$, $$(f\ast ... 0answers 463 views ### Infinite self-convolution for a function I have a mathematical problem that leads me to a particular necessity. I need to calculate the convolution of a function for itself for a certain amount of times. So consider a generic function f : ... 2answers 102 views ### Convolution with sign function I am having some trouble calculating the convolution (f*g)(t) between these two functions:$$ f(t)=e^{-t}1(t) $$where 1(t) is the unit step function, and$$ g(t)=\mathrm{sgn}(t) $$Using ... 2answers 258 views ### Convolution of continuous function with \mathcal{C}^{1} function I'm having difficulty with the following problem: Let f:\mathbb{R}\to\mathbb{R} be continuous on \mathbb{R} such that \mbox{Supp}\left(f\right) is compact and let g:\mathbb{R}\to\mathbb{R} ... 2answers 314 views ### the fourier transform of a “double convolution” Suppose I have a function$$ m(x) = f(x)\int_{-\infty}^{\infty} h(w)g(w-x)dw = f(x)h*g(x)$$I want to find the Fourier transform of m(x) in terms of the Fourier transforms of$f,h,g$but for the ... 0answers 119 views ### When does$|f*g|_{p}=|f|_{1}|g|_{p}$? From Rudin, Real and Complex Analysis, 1st edition, Chapter 7, Problem 4 Suppose$1\le p\le \infty$,$f\in L^{1}(\mathbb{R}^{1})$,$g\in L^{p}(\mathbb{R}^{1})$. Show that the the integral defining ... 0answers 71 views ### bound on Hilbert transform Consider$\widehat{Tf(\xi)}=m(\xi)\hat{f}(\xi)$, where$m(\xi)=(1-\vert\xi\vert)1_{[-1,1]}$, i.e.$T$is the operation of taking Fourier transform and multiplying with the function$m(\xi)$. I am ... 0answers 177 views ### Convolution with a special approximation to the identity function I'm working my way through Stein and Shakarchi's Real Analysis, and I'm having some trouble figuring this exercise out. Given the function$K_\delta$that satisfies the normal approximation to the ... 1answer 257 views ### Convolution converges in infinity norm? Assume$\phi$to be a nonnegative continuous function on the real line with compact support. Also assume that integral of$\phi$over$\mathbb{R}$is normalized to$1$. Let$\phi_e(x) = ...
In a homework question, I was asked to show: (1) in $L^1(R)$, if $f*f = f$, then $f$ must be a zero function. (2) In $L^2(R)$, find a function $f*f=f$. I don't know how to proceed. for (1), $f*f=f$ ...
Here is my example:u and v are the surface measures on the spheres {${x;|x|=a}$} and {${x;|x|=b}$} in $\mathbb{R}^{3}$.Then what's $u\ast v$ ? And what if in $\mathbb{R}^{n}$?