Questions tagged [convolution]
The convolution tag has no usage guidance.
173 questions
3
votes
3
answers
582
views
Approximate identities and pointwise convergence
I'm studying Fourier analysis and have a question about approximate identities.
Let $k_{\epsilon}$ be an approximate identity on $L^{1}(\mathbf{T})$. We know that $k_{\epsilon}*f\to f$ in $L^{1}$ as $...
- 41
3
votes
0
answers
267
views
Link between standard convolution and Day convolution
There is a notion of convolution product between two functors called "Day convolution". (See here nlab for instance) I know that the definition of this notion is inspired by the discrete convolution $$...
- 1,017
59
votes
1
answer
5k
views
Square root of dirac delta function
Is there a measurable function $ f:\mathbb{R}\to \mathbb{R}^+ $ so that $ f*f(x)=1 $ for all $ x\in \mathbb{R} $, i.e $$\int\limits_{-\infty}^{\infty} f(t)f(x-t) dt=1 $$ for all $ x\in \mathbb{R} $.
- 817
1
vote
1
answer
86
views
Subquadratic multiplication of probability mass functions (with log-convolution?)
We are currently looking for a fast, i.e. subquadratic, algorithm for the following equation:
$z_m = \sum_{i,j :\, (i \cdot j) = m} x_i \cdot y_j$.
That is, we are given two finite input vectors $x$ ...
- 13
6
votes
1
answer
909
views
Is there an alternate name for the symplectic convolution?
Looking into the Wigner-Weyl transformation mapping Hilbert space operators to functions on phase-space, I've run up against the need for a symplectic convolution
$$[F\star G](x,p) = \int \!dy\,dk\, ...
- 846
5
votes
2
answers
556
views
Are there multiplicative functions which are not rational?
Vaidyanathaswamy calls an arithmetic function rational if it is the convolution of some finite collection of functions which are either completely multiplicative or inverse to a completely ...
- 9,114
4
votes
0
answers
181
views
Shifted convolution problem for Coefficients of automorphic forms
The shifted convolution problem for coefficients of modular forms is well studied and many estimates were established for the shifted convolution sums of Hecke eigenvalues. So, one may ask about the ...
- 1,257
13
votes
2
answers
610
views
Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?
The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a $C(...
- 4,461
1
vote
2
answers
330
views
Is there a version of the Titchmarsh Convolution theorem to find singular support?
Okay, some terminology, correct me if I'm wrong.
Singular support - the set on which a distribution fails to be smooth. In this case a piecewise function.
Is there a name for $f*f*f$? The ...
- 300
2
votes
1
answer
309
views
Convolution vanishes on an interval
Fix a "test" function $f(x)=x\exp(-x^2)$, which is nonzero except $x=0$. Suppose that $g$ is a function with some necessary regularity. Consider the convolution.
$$
(f\ast g )(x)=\int_{-\infty}^{+\...
- 87
8
votes
0
answers
294
views
Which classes of functions are "convolution ideals"?
If $g$ is continuous then $f*g$ is continuous.
If $g$ is smooth then $f*g$ is smooth.
If $g$ is a polynomial then $f*g$ is a polynomial.
If just one of the two functions belongs to the class of well-...
15
votes
1
answer
604
views
Is the set of the convolutions of two-point measures dense in the set of all measures?
A measure supported in two points is a measure of the form
$$
\mu=\alpha\delta_a+(1-\alpha)\delta_b,
$$
where $a<b$ and $\alpha\in (0,1)$.
The question is:
Given a finite non-negative measure ...
- 321
3
votes
1
answer
556
views
Convolution of measures - entropy growth
Imagine you have two shift-invariant measures $\mu, \nu$ in the Bernoulli space $\{0,1\}^{\mathbb{N}}$ with positive entropy and both are not the Bernoulli measure $(\frac{1}{2},\frac{1}{2})$. I know ...
2
votes
0
answers
2k
views
convolution integral involving modified Bessel functions of the first kind
I'm stuck with this convolution integral ($z \geq 0$)...
\begin{equation}
f_{Z}(z)=\int^{\infty}_{-\infty}f_{1}(x)f_{2}(z-x)dx = \mbox{ } ???
\end{equation}
which represents the pdf of the sum $Z = ...
- 163
3
votes
1
answer
279
views
Complete solution set of a Convolutional Equation?
Here is a problem that am I stuck and I appreciate any help. In essence, I am trying to show that the only solutions for the described problem are the ones provided below. Best..
Setup: In what ...
- 31
0
votes
0
answers
94
views
Do they have the same limit?
Suppose $a(\cdot)\in L^p$ and is symmetric and $b(\cdot)\in L^q$, where $1/p+2/q=2$, $p,q\ge 1$. Consider the quantity $Q_T=$
$$
\frac{1}{T}\int_{\mathbb{R}}dx\int_{[-T,T]^2}d\mathbf{v}\int_{[-T,T]^2}...
- 87
3
votes
1
answer
957
views
Maximum of a mollified/convolution function
I have a function $f:{\mathbb R}\rightarrow {\mathbb R}_+$ which has a unique maximum at $x=0$. $f$ can be symmetric or asymmetric. I am interested on the mollified-f function
$$\tilde{f}(x)=\int_{-\...
- 39
3
votes
0
answers
543
views
Existence and smoothness of convolutions of distributions in Sobolev spaces
Let $f\in H^{s_1}(\mathbb{R}^n)$ and $g\in H^{s_2}(\mathbb{R}^n)$, where $s_1, s_2 \in \mathbb{R}$ and can be positive or negative.
It is easy to show that $f *g$ is defined pointwise when $s_1+s_2\...
- 256
3
votes
0
answers
741
views
Justification of the convolution operation of $L^1(G)$ functions where $G$ is a LCA group (measurability)
Suppose $G$ is a locally compact abelian Hausdorff group (LCA), and $\lambda$ is the Haar measure on it. We all know the convolution of two $L^1(\lambda)$ functions $f$ and $g$ on $G$ is defined as
$$...
- 960
7
votes
3
answers
3k
views
Deconvolution of sum of two random variables
Let $Z = X + c \cdot Y$ where $X$ and $Y$ are independent random variables drawn form the same distribution given by the pdf $g()$ and $0 < c < 1$
I have observations of $Z_i$'s and thus can ...
- 139
3
votes
1
answer
335
views
Bounded convolutions with binomial coefficients
I need to figure out a nice family of decaying functions such that
$\sum_{d=2}^k {k \choose d} f_k(d) \leq 1/k$ and $f_k(d)\geq f_k(d+1)$
How can I figure out what good candidates could be?
Any ...
- 2,537
3
votes
2
answers
423
views
Banach algebra for measures induced by Haar measures
It is classical that $L^1(G, m)$ is a Banach algebra when $G$ is a locally compact group with Haar measure $m$ by using the operation of convolution via the integral
$$(f*g)(y)=\int_Xf(x)g(yx^{-1})\,...
- 1,049
191
votes
34
answers
81k
views
What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...