Questions tagged [convolution]
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164
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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
$$
t_{k} = \sum\limits_{i=0}^{n-k} b_i b_{i+k},
...
- 1,161
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140
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Does convolution with $(1+|x|)^{-n}$ define an operator $L^p(\mathbb R^n) \to L^p(\mathbb R^n)$
Suppose that $f : \mathbb R^n \to \mathbb R^n$ is a locally integrable function. I am interested in the integral
$$ x \to \int_{\mathbb R^n} ( 1 + |y| )^{-n} f(x-y) \;dy $$
If the decay of the ...
- 613
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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 ...
- 43
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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 $g_\sigma$ ...
- 393
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310
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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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146
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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 $\sigma>0$ ...
- 393
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108
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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 ...
- 393
2
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1
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224
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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 $f$ ...
- 393
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94
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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 \...
- 3,113
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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 ...
- 3,113
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190
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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, ...
- 647
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3
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843
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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 $...
- 15.7k
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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 ...
- 203
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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 ...
- 1,188
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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 $\eta$ ...
- 5,735