# 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 ...
• 153
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### 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 ...
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### 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 $\eta$ ...
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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 ...
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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 ...
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### 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 ...
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### 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 ...
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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, ...
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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 ...
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### 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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### 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, ...
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### Is there a name for this type of matrix?

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{...
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I wonder if the convolution $$f(y)=\int_{-\infty}^{+\infty} \mathrm{Airy}(a\cdot x)\cdot e^{-b(y-x)^2} dx$$ can be solved analytically. Or in case not, if there is an ...