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
160 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
788 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
180 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
276 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
158 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
141 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
203 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
107 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
204 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
250 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
702 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
85 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
57 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
520 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
121 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
84 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
65 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
175 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
386 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} ...
0
votes
1answer
5k views

Creating a System Impulse Response in Matlab

Preface: I'm extremely new to Matlab. Ok, so I have a sound file that I loaded in Matlab. Two variables are loaded: ...
0
votes
2answers
815 views

How to calculate a 1D convolution summation?

I hope I said that right. I'm trying to follow along with a convolution example but maybe I am in over my head. I don't understand how in this example they get the values on the right. For example, I ...
1
vote
2answers
116 views

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative.

If $(f \ast g)(n) = \sum\limits_{d|n}f(d)g(\frac{n}{d})$, prove that $\ast$ is commutative. I believe this is convolution. My attempt at the proof: $(f \ast g)(n) = ...
0
votes
3answers
108 views

How does the author get from one step to another?

I have to apply convolution theorem to find the inverse Laplace transform of a given function. I know that convolution is applied when the given function is multiplication of two functions. The ...
3
votes
0answers
154 views

Given a meromorphic function (via its Laurent series), how to obtain the (Taylor series of the) two holomorphic functions it is the quotient of?

Since any meromorphic function $f:\mathbb C\to\mathbb C$ can be expressed as the quotient of two entire functions, i.e. $f(z) = \frac{n(z)}{d(z)}$ where the zeros of the denominator $d(z)$ are $f$'s ...
1
vote
1answer
159 views

How can I write an ordinary function in terms of integral of delta Dirac function?

We have the result $(x * \delta)(t) = x(t)$. Here let $x(t)$ is a real valued function with respect to time and $\delta(t)$ is the unit impulse function. $$ x(t) * \delta (t) = \int_{-\infty}^\infty ...
1
vote
1answer
169 views

Support of Convolution and Smoothing

I just want to know how it follows that $v^{\epsilon} \in C^{\infty}(\bar{V})$? I could see how $v^{\epsilon} \in C^{\infty}(V)$ by using the translations, but I'm having difficulty seeing how it ...
1
vote
1answer
529 views

Integral of repeated convolution of the unit step function

Background Let $\theta$ be the unit step function: $$\theta(x) = \begin{array}{ll} \left\{ \begin{array}{ll} 0 & x \lt 0 \\ 1 & x\ge 0. \end{array}\right. \end{array} $$ Further, the ...
1
vote
0answers
39 views

Finding correlation of data with potentially hidden time lags

Let's say I have few independent variables plus multiple observables that I monitor over time for a system. I'd like to find out if there is any correlation between the observables and any of the ...
5
votes
2answers
328 views

The condition for $Y$ to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$

I would like to know the condition for a random variable $Y$ in order to make $\mathbb{E}[\max\{X_1+Y,X_2\}] > \mathbb{E}[\max\{X_1, X_2\}]$, where $X_1$ and $X_2$ are iid. Any help would be ...
-1
votes
2answers
113 views

Integral of an integral with variable limits

I'd like to prove the following but not sure where to start: ...
1
vote
1answer
242 views

convolve probit function with gaussian [duplicate]

I want to prove the following, however, not sure where to start. $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Where $\Phi(\cdot)$ is the probit function, ...
1
vote
0answers
77 views

LTI system and convolution

I'm reading a rather informal text on (continuous) Linear Time Invariant (LTI) systems. It is just said to be a "black box" that transforms an input signal $x(t)$ into an output signal ...
-4
votes
1answer
258 views

Convolution of Discrete Uniform ,$DU$, Distribution.

If $X\sim DU(k,a,h),\quad -\infty<a<\infty,h>0=1,2,\ldots$ then the probability function is $$P(X=a+jh)=\frac{1}{k},\quad j=0,1,\ldots,k-1$$ Let $Z\sim DU(r,0,s)$ and $Y\sim DU(s,0,1)$ , ...
0
votes
1answer
34 views

How to choose a phase for the deconvolution of an autocorrelation?

Say I have a function, $C=C\left(x\right)$, whose fourier transform is denoted by $c=c\left(k\right)$, i.e. $C\left(x\right)=\sum_{k=-\infty}^{\infty}c\left(k\right)\chi\left(x\right)$, where ...
0
votes
1answer
219 views

Convolution of two dimensional gaussian functions

I want to calculate the sum of two probability density functions. I know that it is: $P_{U+V} (x)= (P_{U} * P_{V})(x)$ If $P_{U}$ and $P_{V}$ are gaussian functions in one dimension, i.e. $P_{U}(x) ...
1
vote
1answer
73 views

Norm of convolution of $n$ Gaussians

If $$f(x)=e^{-(\pi x)^2}$$ and $$\psi_n(x)=(f* f*\dots*f)(x)$$ ($n$ times convolution). Show that $$\lVert \psi_n(x)\rVert = 1$$ (norm in $L^1(\mathbb{R})$). I've tried using the Fourier ...
4
votes
1answer
139 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 ...
1
vote
2answers
107 views

Proof of convolution

I would like to know how I could prove the following convolution: $$ D (f*g) =D f* g =f* Dg $$
3
votes
1answer
1k views

Is there an elementary proof of the convolution theorem?

Is there a way, without using much extra theory (other than the basic ideas used in textbooks deriving the Fourier transform for the first time, and ideally just using general theorems about ...
0
votes
1answer
58 views

Interpret convolution diagram

How do I interpret this "do convolutions" diagram? 1) How are the results computed? 2) When looking at this part: "x[n-k]" Do you interpret convolutions as delays or time reversals? $ y[n]= ...
0
votes
2answers
1k views

Undecimated Wavelet Transform (a trous algorithm) - how to determine 'anchor'/'center' of convolution filter

i am currently implementing the 'Undecimated Wavelet Transform' with the 'a trous' algorithm. See e.g. http://www.znu.ac.ir/data/members/fazli_saeid/DIP/Paper/ISSUE2/04060954_2.pdf, section II-A. As ...
1
vote
2answers
123 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...