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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Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$
It's not difficult to show that
$$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$
On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$. Squaring the first power series and comparing ...
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Convolution Laplace transform
Find the inverse Laplace transform of the giveb function by using the
convolution theorem.
$$F(x) = \frac{s}{(s+1)(s^2+4)}$$
If I use partial fractions I get: $$\frac{s+4}{5(s^2+4)} - \frac{1}{...
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Proving the sum of two independent Cauchy Random Variables is Cauchy
Is there any method to show that the sum of two independent Cauchy random variables is Cauchy? I know that it can be derived using Characteristic Functions, but the point is, I have not yet learnt ...
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Why convolution regularize functions?
There is a tool in mathematics that I have used a lot of times and I'm still not confortable with. In fact I can't figure out (by this I mean that I cannot understand it geometrically) why does ...
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Convolution is uniformly continuous and bounded
Suppose $f\in L^\infty(\mathbb{R})$ and $K\in L^1(\mathbb{R})$ with $\int_\mathbb{R}K(x)dx=1$. Show that the convolution $f\ast K$ is a uniformly continuous and bounded function.
The definition of ...
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Fourier transform as diagonalization of convolution
I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator
$$
A_f(g) = \int f(\tau)g(t-\tau)d\tau
$$
and apply it to $g(t)=e^{ikt}$. ...
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A proof of the fact that the Fourier transform is not surjective from $\mathcal{L}^1(\mathbb{R})$ to $C_0( \mathbb{R})$
Let $f_n = \mathbb 1_{[-n,n]}$ for all $n \in \mathbb{N}$
Compute explicitly $f_n \star f_1$ for all $n \in \mathbb{N}$.
Show that $f_n \star f_1$ is the Fourier transform of $g_n = \frac{
\sin{(2\...
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Finding the distribution of the sum of three independent uniform random variables
Let $X,Y,Z\sim Unif(0,1)$, all independent. Find the distribution of $W=X+Y+Z$
I'm trying to solve this by doing convolution twice. I'm letting $S=X+Y$, then $W=S+Z$. So I should end up with $f_W(w) ...
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What's the difference between convolution and crosscorrelation?
What's the difference between convolution and crosscorrelation? So why do you use '-' for convolution and '+' for crosscorrelation? Why do we need the "time reversal on one of the inputs" when doing ...
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How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function
We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$.
...
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On the closedness of $L^2$ under convolution
It is a direct consequence of Fubini's theorem that if $f,g \in L^1(\mathbb{R})$, then the convolution $f *g$ is well defined almost everywhere and $f*g \in L^1(\mathbb{R})$. Thus, $L^1(\mathbb{R})$ ...
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Convolution of a function with itself
Function $\phi (x)$ is defined as:
$$\phi(x) = \begin{cases} 1 & \text{ if } 0 \leq x \leq 1\\0 & \text{otherwise} \end{cases} $$
How do I find the convolution of $\phi(x)$ with itself? I ...
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Alternating sign Vandermonde convolution
The well-known Vandermonde convolution gives us the closed form $$\sum_{k=0}^n {r\choose k}{s\choose n-k} = {r+s \choose n}$$
For the case $r=s$, it is also known that $$\sum_{k=0}^n (-1)^k {r \choose ...
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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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Density of sum of two uniform random variables
I have two uniform random varibles. $X$ is uniform over $[\frac{1}{2},1]$ and $Y$ is uniform over $[0,1]$. I want to find the density funciton for $Z=X+Y$.
There are many solutions to this on this ...
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