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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16
votes
3answers
2k views

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 ...
16
votes
4answers
4k views

Meaning of convolution?

I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give ...
13
votes
4answers
897 views

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 ...
12
votes
5answers
874 views

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 ...
9
votes
2answers
602 views

Convolution intuition: clarifying Terence Tao's “blurring”/“fuzz” interpretation

On this math.MO post, "What is convolution intuitively?", Terence Tao's answer (in the case where one function is a bump function) involves "blurring" and "fuzz." Could someone clarify his ...
9
votes
1answer
740 views

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 $\alpha$. ...
9
votes
1answer
584 views

The error term in Taylor series and convolution.

I've been wondering a lot why is the remainder of the Taylor expansion of a function, $R_n(x)$, expressed (in one of the many forms) as something very similar to aconvolution. Precisely: $$R_n(x) = ...
8
votes
1answer
939 views

Is this class of periodic functions closed under the (circular) convolution operation ? Help in proving.

I am studying the properties of a particular class of functions, and I'd appreciate some help in proving a property of that class. I started with a class of functions and made some modifications to ...
8
votes
0answers
222 views

Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?

When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
7
votes
4answers
5k views

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 ...
7
votes
2answers
2k views

convolution of a function with itself equals itself

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), $f*f=f$ ...
7
votes
1answer
390 views

ODE Laplace Transforms: what impulse brings an oscillating system to rest?

$2y''+y'+2y=\delta(t-5)$ $y(0)=0, y'(0)=0$ Consider the system given by ODE above in which an oscillation is excited by a unit impulse at $t=5$. Suppose that it is desired to bring the system to ...
7
votes
1answer
161 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
7
votes
1answer
263 views

A Mathematical way to represent a image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image A/input) ...
7
votes
0answers
90 views

Properties of a continued fraction convolution operation

Usually the partial numerators of a continued fraction are all 1s. Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
6
votes
3answers
3k views

Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$

From any textbook on fourier analysis: "It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = ...
6
votes
3answers
899 views

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 ...
6
votes
1answer
140 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) ...
6
votes
2answers
970 views

Convolution of measures

We say that a family of measures $\mu_{t}\to \mu$ weakly if for any $g\in C_{0}$, $\int g d\mu_{t} \to \int g d\mu$. Show that if $\mu_{t}\to \mu$ weakly, then $\nu*\mu_{t}\to \nu*\mu$ weakly, where ...
6
votes
1answer
279 views

Convolution between two distributions

I want to define the convolution $*$ between two distributions $S$ and $T$. For a test function $\varphi$, can I say: $$\langle S * T, \varphi \rangle \doteqdot \langle S, T*\varphi \rangle $$ where ...
5
votes
3answers
260 views

Laplace transforms: Convolution

Find $$1*1*1*\cdots*1\quad n\,\,\text{ factors}$$ that is, a function $f(t)=1$ convolution with itself for a total of $n$ factors. Would anyone mind helping me? I have no idea what I should do. ...
5
votes
2answers
601 views

Limits and convolution

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 ...
5
votes
4answers
2k views

Laplacian 2D kernel - is it separable?

I'm wondering if the 2D laplacian kernel 0 1 0 1 -4 1 0 1 0 is also a separable kernel. How can I find that out?
5
votes
3answers
1k views

Convolution of compactly supported function with a locally integrable function is continuous?

Can someone show me the proof that the convolution of a compactly supported real valued function on $\mathbb{R}$ with a locally integrable function is also continuous? I feel that this is a standard ...
5
votes
2answers
115 views

partially reconstruct information of function convoluted with boxcar kernel

the function (f) I want to reconstruct partially could look like this: The following properties are known: It consists only of alternating plateau (high/low). So the first derivation is zero ...
5
votes
3answers
466 views

if convolution of $f$ with itself remains same, then $f=0$ a.e?

I'm trying to answer the question above.. But I'm not certain in either way. I tried to prove it by giving counter examples.. But it always failed.. Then i also tried to draw contradictions But ...
5
votes
1answer
636 views

On the closedness of $L^2$ under convolution

It is a direct consequence of Fubini's theorem that if $f,g \in L^1(\mathbb{R})$, then the convolution $f *g$ is well defined almost everywhere and $f*g \in L^1(\mathbb{R})$. Thus, $L^1(\mathbb{R})$ ...
5
votes
1answer
421 views

Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?

Let $f$ be a continuous but nowhere differentiable function. Is $f$ convolved with mollifier, a smooth function?
5
votes
2answers
76 views

Is the convolution of a function $f(x)$ and a polynomial $p(x)$ always a polynomial?

After reading the following question: How do I prove a convolution is a polynomial? I want to ask if that is always the expected result, that is to say, does the following holds? A convolution ...
5
votes
2answers
102 views

Easy way to compute $Pr[\sum_{i=1}^t X_i \geq z]$

We have a set of $t$ independent random variables $X_i \sim \mathrm{Bin}(n_i, p_i)$. We know that $$\mathrm{Pr}[X_i \geq z] = \sum_{j=z}^{\infty} { n_i \choose j } p_i^j (1-p_i)^{n_i -j}.$$ But is ...
5
votes
2answers
96 views

Proving the sum of two independent Cauchy Random Variables is Cauchy

Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
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 ...
5
votes
1answer
179 views

Proof of Faà di Bruno's formula using a convolution identity for Bell polynomials?

I have noticed there is an identity for Bell polynomials that can apply of Faà di Bruno's formula. This is a convolution identity that states: $$ (x \ast y)_n = \sum_{j=1}^{n-1} {n \choose j} x_j ...
5
votes
1answer
117 views

Compactness of Convolutions of Compact Measures

This regards measures on $d$-dimensional Euclidean space $\mathbb R^d$ and their associated densities. A super-level set of a density $f : \mathbb R^d \to \mathbb R^+$ at level $t$ is the set $\{x \in ...
4
votes
3answers
471 views

Alternating sign Vandermonde convolution

The well-known Vandermonde convolution gives us the closed form $\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$. For the case $r=s$, it is also known that $\sum_{k=0}^n (-1)^k {r \choose ...
4
votes
2answers
124 views

Solve integral (convolution) equation

Given a function: $u(t) = \exp\left( -\frac{At^2}{1+t}\right),$ $A>0, t>0,$ and an equation: $\frac{d u(t)}{dt} = \int^{t}_0 \phi(t-\tau) u(\tau) d \tau .$ How to find a closed expression for ...
4
votes
3answers
141 views

Convolution doubt

Can someone explain why the general formula of the convolution is this one: $$(f*g)(t)=\int_{-\infty}^{\infty}f(t-\tau)g(\tau)d\tau$$ But when both $f(\tau)$ and $g(\tau)$ are equal to zero for ...
4
votes
1answer
247 views

Conditions for the Convolution $f \ast g$ to be Continuous at a Point

Let $f$ and $g$ be functions on $\mathbb{R}^n$. Let $x_0$ be a given point in the unit ball $B(0,1)$. I am looking for sufficient conditions for the convolution $$ (f \ast g)(x) = \int_{B(0,1)} ...
4
votes
2answers
1k views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
4
votes
2answers
283 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 ( ...
4
votes
1answer
132 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 ...
4
votes
1answer
45 views

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$

Prove that $f\ast g$ is continuous if $f\in C(\mathbb{T})$ and $g\in R(\mathbb{T})$ (Meaning $f$ is continuous and periodic and $g$ is Riemann integrable and periodic). So basically, if we define ...
4
votes
1answer
180 views

Convolution of a probability measure with a smooth function

If $f\in L^1(\mathbb{R}^n)$ and $g\in L^p(\mathbb{R}^n)$ then by Young's convolution inequality we have the estimate: $$ \|f*g\|_{L^p}\leq \|f\|_{L^1}\|g\|_{L^p}.$$ Question: Let $\mu$ be a ...
4
votes
1answer
285 views

Proof that a convolution with $g(x)=\frac{1}{\pi} \frac{r}{r^2+x^2}$ is smooth

It is known that if $g_n: \mathbb{R} \rightarrow \mathbb{R}$, $n=1,2,...$, is in $C_c^{\infty}(\mathbb{R})$, $ \int_\mathbb{R} g_n(x)dx=1$, $supp(g_n) \subset (-r_n,r_n)$,where $0<r_n \rightarrow ...
4
votes
1answer
64 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 ...
4
votes
1answer
125 views

Is there a way to do this with fast convolution?

If you could please offer any advice, this puzzle is driving me mad: I've come across a problem that is trivial to compute in $\mathcal{O}(m^2)$ operations, but which very closely resembles a ...
4
votes
2answers
256 views

Convoluting two surfaces

I'm wondering how the concept of convolution can be extended to 2D. As example, let us take a constant function $z=f(x,y)=1$ with support on $[0..1]^2 \in \mathbb{R}^2$ (see Fig. 1). If we now ...
4
votes
1answer
438 views

Convolution correct?

I want to convolve two square functions (literally square pulses). In my opinion following calculation is correct, but my teacher said, it isn't (I did not yet have the time to speak to him again). Is ...
4
votes
1answer
381 views

“anti-gaussian” 2D convolution kernel

What is the 2D kernel, k, that when convoluted with a 2D signal, f, that is convoluted again with a gaussian 2D kernel, g, produces a result that is closest to the original signal, f'. Something like ...
4
votes
1answer
97 views

A convolution involving binomials

Given $$f(i)\gt0,\:g(i)>0,\:i =0,1,2,3,...\:$$and$$\sum_{i=0}^{\infty}f(i) = 1,\sum_{i=0}^{\infty}g(i) = 1$$Prove that, if$$\frac{g(l-k)f(k)}{\sum_{i=0}^{l}f(i)g(l-i)}=\binom{l}{k}p^k(1-p)^{l-k}\: ...