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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2
votes
1answer
163 views

Fourier transform of convolution for $L^2$ functions

If $f,g\in L^1(\mathbb{R})$, it is not hard to show by definition that $$(\hat{f\ast g)}(t)=\hat{f}(t)\hat{g}(t).$$ But what about if $f,g\in L^2(\mathbb{R})$? The Fourier transform on ...
1
vote
1answer
62 views

Confused with estimator for random variables.

I am working on a practice exercise in preparation for a final this week. I am really stuck on the following problem: Let $X_1, X_2$ be a random sample for a population with the probability density ...
1
vote
1answer
88 views

Any clue how to solve this convolution integral?

With other words: find a (closed) expression for $\;\overline{\mbox{sinc}}(x)$ . $$ \overline{\mbox{sinc}}(x) = \int_{-\infty}^{+\infty} \frac{\sin(\omega\xi)}{\omega\xi} ...
1
vote
0answers
235 views

Approximate convolution of independent Beta variates?

Is there a way to approximate the convolution of Beta variables? Specifically, I am trying to find an approximation to $g(x_0)$: $$g(x_0) = \int \delta(x_0-\sum_{i=1}^{n} a_i x_i) \prod_{i=1}^{n} ...
1
vote
1answer
334 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? ...
3
votes
1answer
158 views

Haar measure, convolution and involutions

I have some problems to follow the proof of the anti commutativity property of the convolution and involution operations defined using a Haar measure as presented in Pedersen's book Analysis Now, ...
1
vote
0answers
40 views

Integral transforms with interesting pointwise multiplications?

The convolution theorem states that the Fourier transform of the convolution of functions equals the pointwise multiplication of Fourier-transformed functions, i.e.: $$\mathcal{F}\{f*g\} = ...
1
vote
1answer
135 views

convolution of measurable function with analytic function

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 ...
4
votes
1answer
68 views

Properties of the operator $T: f\to f*g$

Let g be the characteristic function of [-1/2,1/2]. $T: f\to f*g$ (convolution). I have managed to prove that T is a linear,bounded,self adjoint,injective operator and it's immage is inclused in ...
0
votes
2answers
342 views

What does triple convolution actually look like?

I have to prove associativity of the convolution of three functions. I'm having trouble picturing how the variables will look. For periodic functions $f$, $g$, and $h$, I have the definition $$ (f * ...
1
vote
0answers
239 views

Gamma random variables with fixed sum (different scale parameters)

Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf $Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma ...
2
votes
1answer
45 views

Show $\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = \int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$

I read in an article about Laplace's equation that $$-\int_{\mathbb{R}^n}\Delta_x \Phi(x-y)f(y)dy = -\int_{\mathbb{R}^n}\Delta_y \Phi(x-y)f(y)dy.$$ Could someone explain to me why this is? I ...
2
votes
1answer
60 views

What is the easiest way to find the inverse Laplace of F(s)?

$$ F(s)= \frac{1}{(s-1)^2(1-1/s^2)} $$ Do I have to multiply by $s^2/s^2$ and then use partial fractions or is there a way to use the convolution theorem?
1
vote
1answer
154 views

A self-convolution formula that counts bracket expressions

Problem: Consider an alphabet of size $m+2$, consisting of the two bracket symbols $\ [ \ ] \ $ plus $m$ non-bracket symbols ($m \ge 0$). Define $f_m(n)$ to be the number of length-$n$ strings on this ...
1
vote
2answers
40 views

How do i find the lapalace transorm of this intergral using the convolution theorem?

$$\int_0^{t} e^{-x}\cos x \, dx$$ In the book, the $x$ is written as the greek letter "tau". Anyway, I'm confused about how to deal with this problem because the $f(t)$ is clearly $\cos t$, but ...
4
votes
2answers
330 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 ( ...
0
votes
1answer
49 views

Fourier transform of powers of a function

Assume one has real valued functions $f(x)$ and $g(x)$ that belong to the Schwartz space. I know that the Fourier transforms of $f^3(x)$ and $f^2(x)g(x)$ can be expressed straightforward in terms of ...
1
vote
0answers
54 views

Integral of convolution difference approaches zero

Let $u(x,t)=f(x)\ast\left(\dfrac{1}{2\sqrt{\pi t}}e^{-\dfrac{(at+x)^2}{4t}}\right)$, and suppose that $f\in L^1$. Show that $$\lim_{t\rightarrow 0^+}\int_{-\infty}^\infty|u(x,t)-f(x)|dx=0$$ How ...
2
votes
0answers
197 views

Show that convolution satisfies partial differential equation

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $$u(x,0)=f(x).$$ Let ...
0
votes
1answer
154 views

Convolution of functions and measures

I need some help with this exercise. I'm not sure how to deal with it: Let $f(x)=e^{-x^2}$, $\mu$ the Lebesgue measure in $[0,1]$ and $\nu$ the Lebesgue measure in $[2,\infty)$. I have to find the ...
2
votes
0answers
60 views

Inverse Fourier transform to get convolution

Consider the equation $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x}$$ for a function $u(x,t)$ with initial value $u(x,0)=f(x).$ Let ...
3
votes
1answer
139 views

Bounding for convolution convergence

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
1
vote
1answer
50 views

Convolution convergent in $L^p$

Suppose $f\in L^p(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_t(x)=\dfrac{1}{t}K\left(\dfrac{x}{t}\right)$$ I'm trying to prove that $\lim_{t\rightarrow ...
2
votes
1answer
165 views

Convolution converging uniformly on real line

I'm working on this question and stuck with the following part: Suppose $f\in L^\infty(\mathbb{R})$ and $K,K_1,K_2,\ldots\in L^1(\mathbb{R})$ with $K_n\rightarrow K$ in $L^1$. Why is it true that ...
2
votes
1answer
809 views

Convolution is uniformly continuous and bounded

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function. The definition of ...
2
votes
1answer
181 views

Convolution convergent in $L^\infty$

Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Define $$K_\epsilon(x)=\dfrac{1}{\epsilon}K\left(\dfrac{x}{\epsilon}\right)$$ Is it always true that ...
1
vote
0answers
100 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 ...
3
votes
0answers
62 views

Convolution-like operator on (probability) measures on $[0,1]$ yielding measures on $[0,1]$.

Is there a "correct" or "best" way to define convolution of two (Borel) probability measures on $[0,1]$ to yield another probability measure on $[0,1]$? Recall that the convolution, $\mu * \nu$, of ...
0
votes
1answer
286 views

Probability with bullets and walls

There are two shooters with different guns and bullets. Each shooter shoots a bullet to a different target hanging on a wall. The hit of each bullet follows a normal distribution centered on its ...
1
vote
1answer
165 views

Convolution of distributions is not associative

I need some help with this exercise: It proposes to show that convolution of distributions is not associative: If $T=T_1$ (distribution given by f=1), $S=\delta'$, and $R=T_H$ (we denote as $H$ the ...
1
vote
1answer
82 views

Finding an ideal low pass filter convolution kernel

Let $f \in L^2[-\pi,\pi] $ and let: $$f = \sum_{k=-\infty}^{\infty}\hat{f}(k)e^{ikx}$$ the Fourier expansion of $f$. I want to find a convoultion kernel $g_N$ so that: ...
6
votes
1answer
142 views

Can $f*g = f+g$ for $f$ and $g$ compactly supported?

Let $f$ and $g$ be continuous, compactly-supported functions $\mathbb{R} \to \mathbb{C}$. Can it happen that $f*g = f+g$? Here, $f*g$ denotes the convolution $$(f*g)(s) = \int_\mathbb{R} f(t) g(s-t) ...
0
votes
0answers
207 views

Trying to figure out Fourier transform of {(0.5^n)(u(n))

I'm working in a signals class for continuous signals, and we have this problem shown above. I have tried using this function f_1 X f_2 = F_1 * F_2, where I'm ...
0
votes
1answer
108 views

Using Mollifiers

If we take $f$ to be a smooth function, then how does it follow that we can write $f^{\epsilon}(x)-f(x) = \int_{B(0,1)}\eta(y)(f(x-\epsilon y)-f(x))dy$ where $f^{\epsilon} := \eta_{\epsilon}\ast f$ ...
2
votes
1answer
208 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 $f*\phi_k\to cf$ ...
3
votes
1answer
82 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 ...
0
votes
1answer
67 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 ...
2
votes
1answer
265 views

$L^p$ Spaces, 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 ...
1
vote
1answer
36 views

Convolution composed with an invertible matrix

Let $T$ 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 ...
2
votes
0answers
718 views

How Heaviside step function changes limits of integration

This question involves the Laplace transform of the convolution of two functions. The derivation in my textbook has a step that really confuses me. First I'll lay out their argument. $$ f(t) = f_1(t) ...
0
votes
1answer
56 views

Maximum of one exponential and one uniformly distributed random variable

If X and Y are independent random variables with X exponentially distributed with mean 1 and Y uniformly distributed in [0,1] , how do I find the distribution of Max(X,Y)
2
votes
1answer
86 views

Young's inequality

Let $U \in L^1(\mathbb{R}^d)$ and $\rho \in L^1(\mathbb{R}^d)$ such that $\rho \ge 0$ and the support of $\rho$ is included in $B(0,1)$ (the euclidean unit ball of $\mathbb{R}^d$). Is there a way to ...
1
vote
1answer
58 views

What is the distribution of $X+Y$ where $X \sim U(0,\frac{L}{2})$ and $Y \sim U(\frac{L}{2},L)$?

I started along these lines: Let $Z = X + Y$ where $\frac{L}{2}< z < \frac{3L}{2}$, then, $$f_{X+Y}(z)=f_{Z}(z) = \int f_{X}(x)f_{Y}(z-x)dx$$ However, I am not sure how to fill in the bounds ...
0
votes
1answer
545 views

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 ...
1
vote
2answers
57 views

What is the name of this function similar to convolution?

The functions seems to be very near convolution function, but the only difference is that you integrate by $du$ in convolution, in contrast to $ds$ in this example: $g(t,u) ...
1
vote
1answer
130 views

About the continuity of a convolution product

I need some help with this exercise: If $f\in L_p(\mathbb{R}^n)$ and $g\in L_q(\mathbb{R}^n)$, where $\frac{1}{p}+\frac{1}{q}=1$, Is their convolution $f\ast ...
2
votes
1answer
86 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 ...
0
votes
0answers
67 views

mathematical statistics - convolution and sums

Given integrable functions f and g on $\mathbb{R}$ define $f*g$, the convolution of $f$ and $g$ by $f*g(x) = \int_{-\infty}^\infty f(y)g(x-y) dy$, So i have to show that $M_Z(t) = M_X(T)M_Y(t)$ ...
2
votes
1answer
180 views

Approximation in Sobolev Spaces

Consider the following proof in Lawrence Evans book 'Partial Differential Equations': How does it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in ...
3
votes
1answer
394 views

Evaluating the convolution using the convolution integral

I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where, $$ x(t) = \begin{cases} e^{-at} ...