# Questions tagged [convolution]

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### Convolution of two modified Bessel functions

Does a closed formula (or power series expansion) for the following convolution exist? $$I_{\nu}(x)=\int_{0}^{\infty} K_{\nu}(x-\tau)K_{\nu}(\tau)d\tau$$ Here $K$ stand for the modified Bessel ...
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### 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 ...
1 vote
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### 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 ...
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### 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 $g_\sigma$ ...
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### 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 ...
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### 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 $\sigma>0$ ...
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### 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 ...
206 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 $f$ ...
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### 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 ...
94 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 ...
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### 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 $\eta$ ...
1 vote
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### 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. ...
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### 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 ...
1 vote
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### 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 ...
162 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 ...
173 views

### Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution $$\Gamma: X\longrightarrow X; x\mapsto\int_0^t T(t-s)B(s)x\mathrm{d}s.$$ Here $T(\cdot)$ is a $C_0$-semigroup on some ...
1 vote
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### Special function: Pulse peak modified with a power term

PeakFit (Systat, v. 4.12) is a software for fitting experimental peaks obtained in physics or chemical experiments. Under the miscellenous peak functions, it shows the following equations with a name, ...
1 vote
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### Apply gaussian blur to get original image [closed]

Suppose I have an image A. Is it possible to construct an image A' from A so I can get the ...
133 views

### Local Rankin-Selberg Zeta-function and Coates' p-adic L-Functions

$\DeclareMathOperator\Kern{Kern}\DeclareMathOperator\diag{diag}$ Let $F$ be a non-archimedean local field, $\mathcal{O}$ its ring of integers, $\mathfrak{p}$ its maximal ideal and $\pi$ a uniformizer. ...
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### Subtle distinction in "completeness"?

This is somewhat vague, but please bear with me. Complete metric spaces are supposed to take care of "gaps", they're understood as a natural extension of dense sets. The convolution, defined ...
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I have asked this question a while back on StackExchange but have not received any answer/comment. I received a suggestion to post the same question in here which is more research oriented. Let $k*f(x)... 2 votes 1 answer 99 views ### Equivalent of a local limit theorem in the large deviation region and asymptotics of a convolution operator Let$\{X_i \}_{i \in \mathbb{N}}$be a sequence of i.i.d. random variables satisfying$\mathbb{E} X_1 = 0$and$\mathbb{E} X_1 ^2 < \infty$. Assume that$\{S_n  \}_{n \in \mathbb{N}}$is a non-... 4 votes 1 answer 269 views ### Recovering a function from its Gaussian convolution Let$\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp(-x^2/2)$be the Gaussian density and$f:\mathbb{R}\to\mathbb{R}$another measurable function. Under what conditions can$f$be recovered from its convolution ... 0 votes 0 answers 214 views ### Vector convolution? I am working on a research problem which leads to the following optimization problem: \begin{equation} \hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ... 2 votes 0 answers 79 views ### The square-integrability of$p$and$\nabla u$We consider the stationary Stokes problem in$\mathbb{R}^n$$$\DeclareMathOperator{\Dvg}{\nabla\cdot} \begin{cases} \Delta u + \nabla p = f & \text{ in \mathbb{R}^n} \\ \Dvg u =0. \end{cases} ... 3 votes 0 answers 234 views ### Does convolution by a Schwartz function preserve symbol classes? I am working on a problem involving pseudodifferential operators, and I need a property of the operator "convolution by a Schwartz function". I apologize in advance if the question is ... 2 votes 1 answer 100 views ### Problem regarding vanishing set of convolution Let f vanishes on an open set containing 0. So there exists l>0 such that f vanishes on B(0,2l). So we can choose g\in C_c^\infty (\mathbb{R}^n) (supported on B(0,l)) such that f*g ... 1 vote 0 answers 77 views ### Optimization with convolution in the objective function I would like to minimize the following objective function$$ \| H \ast A - (H \cdot I) \ast B \|_F^2 $$w.r.t. H, where H, I, A, and B are all square matrices of the same size (I is a ... 4 votes 1 answer 204 views ### Why do convoluted convolved Fibonacci numbers pop up from this triangle? Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though T(0,0) ends up different from the OEIS version):$$ T(0,0) = 2;T(1,0) = 1;... 2 votes 1 answer 306 views ### Is$g(v)=\mathbb{E}[f(v+W)]$a differentiable function of$v$when$f$is continuous and$W$is multivariate normal? Suppose$f$is a continuous function on$\mathbb{R}^n$, and$W$has a multivariate normal distribution on$\mathbb{R}^n$. If the expectation $$g(v)=\mathbb{E}[f(v+W)]$$ is defined for all$v \in \... 210 views

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### Can we show that the characteristic function of an infinitely divisible probability measure has no zeros

Let $E$ be a normed $\mathbb R$-vector space, $\mu$ be a probability measure on $\mathcal B(E)$ and $\varphi_\mu$ denote the characteristic function$^1$ of $\mu$. Assume $\mu$ is infinitely divisible, ...
For my thesis in neural networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:  \begin{bmatrix} a&b&c\\ d&0&-d\\ -c&-b&-a \end{...