Questions tagged [convolution]
Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.
3,032
questions
99
votes
4
answers
13k
views
Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
It's not difficult to show that
$$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$
On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
- 1,520
36
votes
6
answers
16k
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 ...
- 22.7k
30
votes
4
answers
1k
views
Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
- 321
29
votes
3
answers
2k
views
Is the rectangular function a convolution of $L^1$ functions?
Do there exist functions $f,g$ in $L^1(\mathbf{R})$ such that the convolution $f \star g$ is (almost everywhere) equal to the indicator function of the interval $[0,1]$ ?
- 447
28
votes
3
answers
34k
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 $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = \int_{-\...
- 633
27
votes
3
answers
10k
views
Meaning of convolution?
I am currently learning about the concept of convolution between two functions in my university course. The course notes are vague about what convolution is, so I was wondering if anyone could give ...
- 335
26
votes
1
answer
726
views
Find $f$ such that $f \star f(x) = \frac{1}{1-x}$.
I'm looking for a measurable function $f$ defined on $]0,1[$ such that :
$$f \star f(x) = \int_{0}^1 f(x-y) f(y) \ \mathrm{d}y = \frac{1}{1-x}$$
for (almost) any $x \in ]0,1[$.
Is it possible to find ...
- 2,190
25
votes
2
answers
17k
views
Proving the sum of two independent Cauchy Random Variables is Cauchy
Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
- 13k
23
votes
6
answers
27k
views
Can someone intuitively explain what the convolution integral is?
I'm having a hard time understanding how the convolution integral works (for Laplace transforms of two functions multiplied together) and was hoping someone could clear the topic up or link to sources ...
- 575
22
votes
1
answer
41k
views
Proof of associativity of convolution
I intend to prove the associativity of convolution but failed after several trials, i.e.
$(f \ast g) \ast h = f \ast (g \ast h)$
where $(f \ast g)(t) = \int^{t}_{0}f(s)g(t-s)ds $
There are a number ...
- 341
20
votes
2
answers
9k
views
Fourier transform as diagonalization of convolution
I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator
$$
A_f(g) = \int f(\tau)g(t-\tau)d\tau
$$
and apply it to $g(t)=e^{ikt}$. ...
- 403
19
votes
2
answers
4k
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 $\alpha$.
...
- 28.1k
17
votes
5
answers
4k
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 ...
- 20.9k
17
votes
2
answers
20k
views
Is the convolution an invertible operation?
I have a signal $f(x,y)$, which is discrete. I convolve this signal with a kernel $h(x,y)$:
$y(x,y) = f(x,y) \star h(x,y)$ (where $\star$ is the convolution operator)
Can I obtain $f(x,y)$ given only $...
- 320
15
votes
1
answer
6k
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, $L^1(\mathbb{R})$ ...
- 3,888