Questions tagged [convolution]
The convolution tag has no summary.
178 questions
3 votes
0 answers
144 views
Solving an SDP that involves a decreasing sequence of circulant and skew-circulant matrices
Question. I would like to find an analytic solution of the following semidefinite program, where $e_0 = (1, 0, \ldots, 0)^{\top}$. $$ \begin{aligned} y = \min &\frac{1}{n} \sum_{i,j=0}^{n-1} A[i,j]...
- 81
1 vote
0 answers
125 views
Young's Inequality for Hardy spaces
Let $d\ge 1$ be an integer, let $\mathcal H^1(\mathbb R^d)$ be the Hardy space on $\mathbb R^d$ and $L^{d,\infty}(\mathbb R^d)$ be the Lorentz space of index $(d,\infty)$ on $\mathbb R^d$. Is it true ...
- 16.6k
1 vote
0 answers
61 views
Analytical expression for the transverse overlap of two uniform spheres
I would like to prove the following formula for the transverse overlap of two uniform spheres: $$ I(\mathbf{b}) = \int d^2\mathbf{s} \; n(\mathbf{s}) \, n(\mathbf{b} - \mathbf{s}) \\ = \left( \frac{...
- 29
4 votes
1 answer
167 views
Regularization of vector fields (convolution, semigroups...)
Let $M, N$ be smooth Riemannian manifolds. We consider a map $A : M \times N \rightarrow TM$ such that for all $z \in N, A(\cdot, z)$ is a vector field on $M$. Let us suppose that $A$ is smooth in the ...
- 115
4 votes
1 answer
318 views
Bounds on Sobolev norms of mollified functions - interpolation
Let $a,b\in \mathbb{R}$ with $a< b$, $n\in \mathbb{N}$ with $n>0$, $1\le p\le \infty$, let $W^{a,p}(\mathbb{R}^n)$ be the corresponding Sobolev space, and let $(m_{\delta})_{\delta>0}$ be a ...
- 57
0 votes
0 answers
111 views
Can Parseval's theorem be extended to the "nested" Fourier series representation of $f(x)$?
This is a cross-post of this question I posted on Math StackExchange a couple of months ago that has not yet received any answers or even comments (other than a single comment of my own). Assuming ...
- 1,221
0 votes
0 answers
40 views
When can the action of a large convolution kernel be replicated by a sequence of convolutions with smaller kernels?
Suppose I have some target filter $f(x)$ supported on the $r$-ball. I am curious what the necessary/sufficient conditions are for when the convolution operator $f \star h$ can be computed via a ...
- 204
12 votes
3 answers
439 views
Probability that a random variable is smaller than its expectation along the CLT
Let $X$ be a random variable with $0$ mean. I'm interested in evaluating the probability $p_1$ that $X$ is below its expectation. $$p_1=\mathbb{P}(X\leq 0)$$ Furthermore, we define for $n\geq1$ $$p_n =...
- 323
2 votes
0 answers
113 views
Can we write this function as a convolution product?
Let $\lambda_{sym^{r}f}(n)$ be the $n-$th coefficient in the Dirichlet series representing the symmetric power $L-$function attached to a primitive form $f$ of weight $k$ and level $N$. It is known ...
- 1,297
0 votes
0 answers
132 views
Derivative bounds for self convolution of the spherical measure in $R^d$
While reading this article on near $L^1$ estimates for the spherical lacunary maximal function, I came across the estimate $$ |\partial^{\gamma} (\widetilde{\sigma} \ast \sigma)(x)| \lesssim |x|^{-(1 +...
- 1
3 votes
1 answer
532 views
Approximate square root of Dirac delta function on $\mathrm{SL}_2(\mathbb{R})$
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\AdS{AdS}$I hope to find a sequence of complex-valued functions $\{f_i(g)\}$ on the group element $g$ of a locally compact group $\SL(2,\mathbb{R})$ so ...
- 175
1 vote
0 answers
104 views
Breakdown of the fourier series identity $e_n e_m = e_{n+m}$ on a perturbed torus
I would like to apologize for leaving things a bit vague. I think my question could be stated much more precisely but right now that is difficult for me to do. I nevertheless think it is an ...
- 389
2 votes
0 answers
99 views
On a possible generalization of heat kernel semigroups on Lie groups
Let $G$ be a compact matrix Lie group with Haar measure $\mu$. Then the heat kernel $\rho: G\times (0,\infty) \rightarrow \mathbb{R}$ satisfies (1) $\rho(g_1g_2,t)=\rho(g_2g_1,t)$ for all positive $t$,...
- 617
1 vote
0 answers
97 views
regularity convolution of a $L^2$ function with $W^{1,1}$ function [closed]
Let $u\in L^2(\mathbb R)$ and $w \in W^{1,1}(\mathbb R)$, we consider the convolution $$u*v$$ Is it true that $w*u \in W^{1,2}(\mathbb R)$? What regularity can we put on $w$ for this to be true?
- 73
6 votes
1 answer
399 views
Integral convolution equation $\int_{B_n(R) } e^{- \| x - t\|} d\nu(t) = e^{- \|x \|^2/2}$ on $x \in B_n(R)$. Find measure $\nu$
Let $B_n(R)$ denote the $n$ ball centered at zero with radius $R$. We are interested in the following integral equation: given $R>0$ and $\lambda>0$, let \begin{align} \int_{B_n(R)} e^{- \...
- 671
6 votes
0 answers
372 views
Distribution class closed under convolution counterexample?
Define the class of probability density functions $\mathcal{C}$: $\,p \in \mathcal{C}$ iff $p(x)=p(|x|)$, and $\log p(\!\sqrt{x})$ is convex on $[0,\infty)$. Conjecture: if $p,q \in \mathcal{C}$, then ...
- 401
3 votes
0 answers
125 views
Is it true that p-integrable function can be written as a convolution of an integrable function and p-integrable function?
We know that convolution of an integrable function with an $p$-integrable is an $p$-integrable function. This follows from Young's inequality. My question: Is it true that $L^p(\mathbb{R}^n)\subseteq ...
- 31
6 votes
1 answer
509 views
Analyticity of $f*g$ with $f$ and $g$ smooth on $\mathbb{R}$ and analytic on $\mathbb{R}^*$
Suppose that we have two real functions $f$ and $g$ both belonging to $\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$ analytic on $\mathbb{R}\setminus\{0\}$ but non-analytic at $x=0$. Is the convolution (...
- 403
1 vote
0 answers
163 views
Fourier transform relation for spherical convolution
Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$. The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \...
- 2,602
1 vote
0 answers
117 views
Is there an generalisation of convolution theorem to integral transforms
Basic convolutions can be computed efficiently by taking fourier transforms and applying the convolution theorem. Is there something analogous for a more general transform, where we have a varying ...
0 votes
0 answers
92 views
Probability distribution of total time for a job, given a workflow graph
$$ \begin{array}{cccccccccccc} & & \text{A} \\ & \swarrow & & \searrow \\ \text{B} & & & & \text{C} \\ & \searrow & & \swarrow \\ \downarrow & &...
1 vote
1 answer
189 views
Small total variation distance between sums of random variables in finite Abelian group implies close to uniform?
Let $\mathbb{G} = \mathbb{Z}/p\mathbb{Z}$ (where $p$ is a prime). Let $X,Y,Z$ be independent random variables in $\mathbb G$. For a small $\epsilon$ we have $\operatorname{dist}_{TV}(X+Y,Z+Y)<\...
- 23
1 vote
1 answer
108 views
Lower bound the best $\alpha$-Hölder constant of a convolution
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 1,163
1 vote
2 answers
139 views
Is the difference between $\alpha$-Hölder constants of $f*\rho$ and $g*\rho$ controlled by $\|f-g\|_\infty$?
Let $\mathcal D_1$ be the set of bounded probability density functions on $\mathbb R^d$. This means $f \in \mathcal D_1$ if and only if $f$ is non-negative measurable such that $\int_{\mathbb R^d} f (...
- 1,163
1 vote
1 answer
171 views
Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
- 424
1 vote
0 answers
319 views
Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$
$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
- 1,221
4 votes
1 answer
459 views
Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution
Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$. Question. What are necessary and sufficient conditions on $Q$ to ensure ...
- 7,033
1 vote
1 answer
182 views
Inequality with convolution
I have some troubles with the following problem: A definition Let $\sigma_1$ and $\sigma_2$ two positive numbers. We denote for all $x\in\mathbb{R}$, >$G_\sigma\left[ \phi \right](x)$ the gaussian ...
- 403
1 vote
1 answer
101 views
Can non-periodic discrete auto-correlation be inversed?
I'm trying to understand whether discrete auto-correlation can be reversed. That is, we are given $t_0, \dots, t_n \in \mathbb C$ and a set of equations $$ t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k}, ...
- 1,211
0 votes
1 answer
242 views
Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral $$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$ If the decay of the ...
- 641
1 vote
0 answers
49 views
Spectrum of the convolution of the Maxwell collision kernel with a distribution
Given the Maxwell collision kernel $A(z) = |z|^2I_d - z \otimes z$, where $I$ denotes the $d\times d$ identity matrix and $z\otimes z = zz^T$ is the outer product, it is easy to see that $A(z)$ has ...
- 43
0 votes
0 answers
97 views
Lower bound of the derivative $(f*g_\sigma)'$ at the zero-crossing point
I am stuck with the following problem. Let consider $f$ a smooth real function such that: $f$ is negative before 0, $f$ is positive after 0, we have $|f'(0)|>0$. Let $\sigma>0$ and $g_\sigma$ ...
- 403
5 votes
1 answer
628 views
Recent progress restriction conjecture - Problem 2.7 (Terence Tao lecture notes)
I've been tackling the following problem for some time, Problem 2.7. (a) Let $S:=\left\{(x, y) \in \mathbf{R}_{+} \times \mathbf{R}_{+}: x^2+y^2=1\right\}$ be a quarter-circle. Let $R \geq 1$, and ...
3 votes
0 answers
185 views
Inequality involving convolution roots
I am struggling with the following problem. Let $f$ be a real smooth function. Let assume that $f$ is: increasing strictly convex on $(-\infty,0)$ strictly concave on $(0,+\infty)$ Let $\sigma>0$ ...
- 403
2 votes
1 answer
149 views
Uniqueness of the zero of $f-f*G_\sigma$ with $f$ convex/concave
I am struggling with the following problem. Let $f$ be a real smooth function: strictly convex on $(-\infty,0)$, strictly concave on $(0,\infty)$, strictly increasing. For $\sigma>0$, how can one ...
- 403
2 votes
1 answer
290 views
Distance between root of $f$ and its Gaussian convolution
Let $f$ be a : $f\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R})$, for all $x> 0,~f(x)>0$, for all $x< 0,~f(x)<0$, I am struggling to find a bound for the distance between the root of $f$ ...
- 403
0 votes
0 answers
138 views
Does the tensor product of mollifiers work for $L^{p,q}$ spaces?
Let $X$ and $Y$ be compact regions of $n$- and $m$-dimensional Euclidean spaces respectively. For any $p,q \in [1,\infty)$, define $L^{p,q}(X \times Y)$ be the space of real valued functions $f :X \...
- 3,745
1 vote
1 answer
193 views
Convolution with the Jacobi Theta-function on "both the space and time variables" - still jointly smooth?
Let $\Theta(x,t)$ be the Jacobi-Theta function: \begin{equation} \Theta(x,t):=1+\sum_{n=1}^\infty e^{-\pi n^2 t} \cos(2\pi n x) \end{equation} Usually, the heat equation with the periodic boundary ...
- 3,745
3 votes
1 answer
294 views
Is there a real/functional analytic proof of Cramér–Lévy theorem?
In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, ...
- 657
6 votes
3 answers
1k views
Convolution of $L^2$ functions
Let $u\in L^2(\mathbb R^n)$: then $u\ast u$ is a bounded continuous function. Let me assume now that $u\ast u$ is compactly supported. Is there anything relevant that could be said on the support of $...
- 16.6k
4 votes
1 answer
305 views
Just how regular are the sample paths of 1D white noise smoothed with a Gaussian kernel?
Adapted from math stack exchange. Background: the prototypical example of---and way to generate---smooth noise is by convolving a one-dimensional white noise process with a Gaussian kernel. My ...
- 203
4 votes
0 answers
146 views
Convolution algebra of a simplicial set
Consider a simplicial set $X^\bullet$ with face maps $d_i$ (assume the set is finite in each degree so there are no measure issues). Then given two functions $f,g:X^1\to \mathbb{C}$ one can form their ...
- 1,187
0 votes
1 answer
222 views
Does convolution commute with Lebesgue–Stieltjes integration?
Let $g: \mathbb R \to \mathbb R$ be a function of locally bounded variation, and $f$ a locally integrable function with respect to $dg$, the Lebesgue–Stieltjes measure associated with $g$. Let $\eta$ ...
- 9,448
1 vote
1 answer
611 views
Extracting eigenvalues of a circulant matrix using discrete Fourier matrix
The eigenvalues of a circulant matrix $C$ can be extracted as $$ \Lambda=F^{-1} C F $$ where the $F$ matrix is a discrete Fourier transform matrix and $\Lambda$ is a diagonal matrix of eigenvalues. ...
- 873
5 votes
0 answers
188 views
Computing sums with linear conditions quickly
Let $f:\{1,\dotsc,N\}\to \mathbb{C}$, $\beta:\{1,\dotsc,N\}\to [0,1]$ be given by tables (or, what is basically the same, assume their values can be computed in constant time). For $0\leq \gamma_0\leq ...
- 21.7k
2 votes
0 answers
139 views
A technical question concerning convolution product
Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$. Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so ...
- 1,155
3 votes
1 answer
199 views
Convolution between normal distribution and the maximum over $m$ Gaussian draws
$\DeclareMathOperator\erf{erf}$ Let's consider the Gaussian distribution $P_X(x)= \frac{1}{\sqrt{2 \pi \sigma^2}} e^{- \frac{x^2}{2 \sigma^2}}$. Now consider the random variable $W \equiv \max \{ X_1, ...
- 321
3 votes
0 answers
171 views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a ...
- 299
1 vote
0 answers
94 views
Solve linear matrix equation involving convolution
I am facing following equation: $$ A * X + C \cdot X = D $$ with: $A, C, D \in \mathbb{R}^{n \times n}$ some known matrices without any particular structure, $X \in \mathbb{R}^{n \times n}$ the ...
- 11
2 votes
1 answer
232 views
Approximating a function by a convolution of given function?
Let $g:\mathbb{R}\to \mathbb{R}$ be a given differentiable function of exponential decay on both sides. Now let us be given a function $f:\mathbb{R}\to \mathbb{R}$, also of exponential decay, if you ...
- 21.7k