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
922 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 ...
13
votes
4answers
694 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 ...
13
votes
4answers
3k 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 ...
12
votes
5answers
797 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
1answer
494 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
506 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 $\alpha$. ...
8
votes
0answers
209 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
2answers
387 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 ...
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
223 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
1answer
269 views

ODE Laplace Transform an impulse bring 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
893 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 ...
7
votes
0answers
83 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
735 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
130 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
780 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
142 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 ...
6
votes
1answer
228 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
241 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
3answers
1k 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
2answers
517 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
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
96 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
296 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
429 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
366 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
318 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
59 views

Intuitive basis of Mobius inversion?

If we're given $f(n)= \sum_{d|n}g\left(\frac{n}{d}\right),n \in \mathbb{N},$ then Mobius inversion gives $$g(n)=\sum_{d|n}\mu \left( d\right) f \left( \frac{n}{d}\right).$$ Also, the generalised ...
5
votes
1answer
108 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
3k 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 ...
4
votes
3answers
361 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
1answer
215 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
54 views

Does a convolution of a function $f(x)$ and a polynomial $p(x)$ will always result in 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 ...
4
votes
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 ( ...
4
votes
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 ...
4
votes
1answer
101 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
263 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
2answers
93 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 ...
4
votes
2answers
351 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
1answer
57 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
120 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
221 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
369 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
357 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
2answers
146 views

How to make 3D object smooth?

I want to make the below picture into an egg with smooth surface. For the implementation in Mathematica, please, see this thread here. This thread considers mathematical methods to achieve the goal ...
4
votes
1answer
246 views

How can I compute $\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$

If $f(x)=\text{arccot}(x)$ for non-negative $x$ and $0$ otherwise, how can I calculate $$\int_{-\infty}^\infty f(x)f(y-x)\, \mathrm dx$$ for $y\in\mathbb{R}$?
4
votes
1answer
61 views

What is the pdf of $Z=X/\max(X,Y)$ with $X,Y$ exponentials of lambda parameter?

Given $X,Y$ 2 independent r.v.'s both distributed as $\exp(λ)$, what is the pdf of $Z=X/\max(X,Y)$?
4
votes
0answers
98 views

Symbolic math engines barf on this ostensibly tractable integral.

$$\frac14 \int_{-M\pi}^{N\pi - s} \cos(tu/M) \cos((t+s)u/M)(1-\cos(t/M))(1-\cos((t+s)/N))\space \mathrm d t$$ with integer $u$. Alpha runs out of time. Maxima gives a tremendous result that can ...
4
votes
1answer
44 views

Can FFT be adapted for deconvolution of non-periodic functions?

Can a non-periodic function be padded at the boundaries and deconvolved with inverse FFT? Since a Toeplitz matrix can be embedded in a circulant matrix to perform the deconvolution, is there an ...
4
votes
0answers
100 views

Are any of those quotient rings isomorphic to other well known rings?

(1) Let $C_b(\mathbb{R})$ be the ring (with pointwise multiplication and addition) of bounded continuous functions. Let $I_0=\{f_{(x)} \in C_b(\mathbb{R}) \space | \space lim_{x \to \pm ...