Questions tagged [convolution]
Questions on the (continuous or discrete) convolution of two functions. It can also be used for questions about convolution of distributions (in the Schwartz's sense) or measures.
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Distribution of the sum of absolutes values of T-distributed random variables
Where $X$ is a r.v. following a symmetric $T$ distribution with $0 $mean and tail parameter $\alpha$.
I am looking for the distribution of the $n$-summed independent variables $ \sum_{1 \leq i \leq n}|...
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What plays the role of the identity for the generalized convolution associated to the Fourier-Bessel transform?
In traditional Fourier theory, the Dirac delta plays the role of an "identity" for the $L^1$ algebra with respect to the usual convolution. The convolution is traditionally built out of group ...
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Convolution of a box function over the hypercube
This looks like a simple convolution over the $p$-dimensional hypercube, but I am unable to find a closed form expression for arbitrary integer $p$:
$$f_p(a)=\int_0^1 dx_1\int_0^1 dx_2\cdots \int_0^1 ...
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Is there a closed form solution of $f(x)^2+(g*f)(x)+h(x)=0$ for $f(x)$?
When $g(x)$ and $h(x)$ are given functions, can $f(x)^2+(g*f)(x)+h(x)=0$ be solved for $f(x)$ in closed form (at least with some restrictions to $g,h$)? (The $*$ is not a typo, it really means ...
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Integrating $\int_{0}^{\infty} \frac{p^6 dp }{1 + a p^4 + b p^6 } \int_{0}^{\pi}\frac{\sin^5 \theta \,d\theta}{1 + a |p-k|^4 + b |p-k|^6 }$
This is my first question here, so I hope I'm not giving too little/too much information. I need some help calculating (or even approximating) an integral which I've been wrestling with for a while.
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Properties of a continued fraction convolution operation
Usually the partial numerators of a continued fraction are all 1s.
Has anyone considered the operation where you convolve 1 continued fraction with another, in other words, make a new continued ...
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Potential uses for viewing discrete wavelets constructed by filter banks as hierarchical random walks.
I have some weak memory that some sources I have encountered a long time ago make some connection between random walks and wavelets, but I am quite sure it is not in the same sense.
What I was ...
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Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
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Evaluate $\int_0^t e^{-\lambda s} \,\text{erf}(\ln(t-s))\,\text{d}s$?
I'm trying to efficiently graph the function $I(t)$ for $t>0$ where
$$I(t) :=\int_0^t e^{-\lambda s} \,\text{erf}(\ln(t-s))\,\text{d}s,\qquad \lambda>0$$
but its evaluation is beyond my powers ...
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Positivity of a convolution integral
Let $f\in\mathcal{S}(\mathbb{R}^3)$ a real function. Consider the following integral
$$I_f:=\int_{\mathbb{R}^3}\int_{\mathbb{R}^3}\frac{f(x)f(y)}{|x-y|^2}dydx$$
Observe that, either if $f$ is ...
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Gamma random variables with fixed sum (different scale parameters)
Given a vector of independent random variables $\{X_i\}_{i=1..N}$, each of which is distributed according to a Gamma-distribution with pdf
$Pr(X_i=x;\alpha_i,\beta_i) = \frac{1}{\Gamma (\alpha_i)}\...
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Conditions for $P(X_1 + \cdots X_{n+1} = 0) < P(X_1 + \cdots X_n = 0) $ to hold for all $n$?
Let $X_i$ be iid random discrete variables with pmf $f$ .
We may restrict ourselve to pmfs with finite even support: $f \in G_N$ ($ f[k] > 0 \implies |k| \le N$) or perhaps $f \in G^{+}_N$ ($ f[k] &...
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Show that any non-trivial ideal of $(L_1,*)$ is dense
This is related to this other question, I mean, the linked question comes to my mind trying to solve the following exercise:
Show that any non-trivial ideal of $(L_1,*)$ is dense.
Here $(L_1,*)$ ...
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Orthogonality of Periodic Sinc function
Background
The Weyl-Heisenberg set $\mathcal{S}$ of sinc functions forms an orthonormal basis for $L_2(R)$, where we have
$$\mathcal{S}=\{s_{km}(t)=\frac{sin(\pi (t-m)}{\pi (t-m)}\exp(j2\pi kt)|m,k\...
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Numerical stability of Winograd short convolution algorithm
Similar to how Strassen matrix multiplication is an asymptotically faster matrix-multiplication algorithm, there exists a similar idea for convolution by (short) filters called Winograd convolution [1,...
