partial differential equations – How to prove the convolution of Newtonian Potential and a function for n=1?

I’m searching for a way to prove the Newtonian potential, given as a convolution in the form of:

$$ (E * Delta f)(x) = int_R E(x-y) Delta f(y) d^ny = f(x)$$

where $n=1$. Apparently this proof is supposed to be easy, but the only idea I have had is to use $x_o$ as the value and try and prove the formula $int E(x_o -y)Delta f(y) dy = f(x_o) = 0$, but really no clue if that’s a good way of doing it.

Any help on how to prove something like would be helpful! I’m quite confused as to how the $x-y$ part of the integral is supposed to be solved. I’m also not clear on how the E is supposed to be integrated.

Thanks!

convolution – Is there a name for this type of matrices?

For my thesis in Neural Networks, I was trying to find a way to generalize a Sobel operator. I quickly thought of this:
$$
begin{bmatrix}
a&b&c\
d&0&-d\
-c&-b&-a
end{bmatrix}
$$

For example here is a quick list of different Sobel operators:
$$
begin{matrix}
& a & b & c & d \
text{Vertical Sobel} & 1 & 2 & 1 & 0 \
text{Horizontal Sobel} & 1 & 0 & -1 & 2 \
text{Diagonal Sobel} & 2 & 1 & 0 & 1 \
text{Anti-Diagonal Sobel} & 0 & 1 & 2 & -1 \
end{matrix}
$$

I wanted to give a name to this kind of matrices so that I can reference it later throughout my thesis. I was thinking of calling them Antisymmetric, but I’ve seen that that term is also used for skew-symmetric matrices. How would you call them? Derivative Matrices? General Sobel Matrices?

integration – Convolution of an Airy function with a Gaussian

I wonder if the convolution

begin{equation}
f(y)=int_{-infty}^{+infty} mathrm{Airy}(acdot x)cdot e^{-b(y-x)^2} dx
end{equation}

can be solved analytically. Or in case not, if there is an analytic expression for the zeros of $f(y) = 0$ and for which values of $a$ and $b$ they exist.

edit: $a>0$ and $b>0$

real analysis – Scaling of double convolution

I am interested in the scaling of

$$F(x_1,x_4)=int_{mathbb R^2} e^{-vert x_1 -x_2 vert -varepsilon vert x_2 -x_3 vert- vert x_3 -x_4 vert } dx_2 dx_3 $$

In particular, I suspect that

$$F(x_1,x_2) le C varepsilon^{-n} e^{{varepsilon}vert x_1 -x_4vert}$$
for some universal $C>0$ and $n ge 0$.

But this is really only based on pure heuristic and I do not know which $n$ could be optimal here.

Cancelling a term by convolution

I am trying to cancel one term within a function by convolution.

y[n] = x[n] + 2*x[n-N]

y[n] * h[n] = x[n] 

how can I find h[n] such that convolution with h[n] cancels the second term of y[n] but leaves the first?

Thank you

real analysis – Supremum norm for convolution in sequence spaces

Question:

Suppose that $1 leq p leq infty$, and the convolution $x ast y$ exists. For sequences $x in ell^p(mathbb{Z})$ and $y in ell^q(mathbb{Z})$, we have

$$||x ast y ||_{infty} leq ||x||_p||y||_q,$$

and $x ast y in ell^{infty}$.


We know $p=1$ and $p=infty$ follow by Young’s Inequality ($||x ast y||_p leq ||x||_p||y||_1$).

For $1 < p < infty$, I’m trying take the supremum over what Hölder’s inequality gives. We can define the $n$-th “convolution” as

$$(x ast y)_n = sum_{i=-infty}^{infty} x_iy_{n-i},$$

and so Hölder’s gives

$$||(x ast y)_n||_1 leq ||x||_pleft(sum_{i=-infty}^{infty} |y_{n-i}|^qright)^{frac{1}{q}}.$$

Taking the sup over $n$,

$$||(x ast y)_n||_{infty} leq ||x||_p sup_{n in mathbb{Z}} left(sum_{i = -infty}^{infty} |y_{n-i}|^qright)^{frac{1}{q}} stackrel{?}{=} ||x||_p||y||_q.$$

Can I just reindex this in some manner, $i mapsto n – i$? Is this equivalent to the translation invariant that one would do for the convolution in function spaces?

Hopf “algebroid” structure of a groupoid convolution algebra?

This question is already posted in math.stackexchange, but didn’t receive any answer. I’m not sure if this question fits in here, but surely someone in here can guide me to the correct answer.

To male thinks simple as possible, lets say we have a discrete group $G.$ Then the then the group algebra $mathbb{C}(G)$ (of finitely supported complex valued functions on $G$) has a convolution and an involution operation given by $$(fstar g)(x)=sum_{x=ab}f(a)g(b), qquad f^{ast}(x)=overline{f(x^{-1})}$$

It is easier to interpret $mathbb{C}(G)$ as the free complex vector space spanned by $G$ for notational convenience. I came across a statement that says this $ast$-convolution algebra has a natural Hopf algebra structure given by comultiplication $Delta(g)=gotimes g$ and counit $epsilon(g)=1,$ then extended linearly. Also antipode is given by $ast$-operation extended antilinearly.

Now I would like to know, what happen if we replace the group $G$ with a groupoid? My naïve guess is that we would get a Hopf algebroid (many object analogue of the known construction). If it is the case, how would the coalgebra look like? Can anyone explain me this structure or direct me to a (simple) reference?

real analysis – $1$-dimensional Fourier transform of $n$-dimensional convolution

The $n$-dimensional Fourier convolution theorem says that for any two (suitable) complex-valued functions $f$ and $g$ which are defined on $mathbb{R}^n$, one has
$$mathcal{F}_n(f*g)(xi)=mathcal{F}_nf(xi)mathcal{F}_ng(xi),,,,,,,,,forall xi inmathbb{R}^n,$$
where $mathcal{F}_n$ denotes the $n$-dimensional Fourier transform and $*$ the convolution.

Question : Now, take any two (suitable) complex-valued functions $f$ and $g$ which are defined on $mathbb{R}^n$, and defines
$$bar{f*g} : xiinmathbb{R} longmapsto (f*g)(xi,xi,dots,xi)$$
(embeds $mathbb{R}$ diagonally into $mathbb{R}^n$), then what can we say about the $1$-dimensional Fourier transform of $bar{f*g}$ ? i.e. what is
$$mathcal{F}_1(bar{f*g})(xi) ?$$

I have tried to follow the proof of Fourier convolution theorem, but it seems to me that this lead us to nothing, even in the case $n=2$.. Can someone shed some light here ?

fa.functional analysis – Laguerre convolution truncation error

Suppose i have extended two d-variate functions $f$ and $g$ (two densities: positives and integrate to one) supported on $mathbb{R}_{+}^d$ into the following (tensorised) Laguerre($alpha = 0$) orthonormal basis of $L^2(mathbb R_{+}^d)$: $$left(varphi_{mathbf k}(mathbf x) = sqrt{2}^d e^{-lvert mathbf x rvert} sumlimits_{mathbf j le mathbf k} binom{mathbf k}{mathbf j} frac{(-2mathbf x)^{mathbf j}}{mathbf j !}right)_{mathbf k in mathbb N^d}.$$

I obtain coefficients $a_{mathbf p}$ for $f$ and $b_{mathbf p}$ for g, such that :

$$f(x) = sumlimits_{mathbf p in mathbb{N}^d} a_{mathbf p} varphi_{mathbf p}(mathbf x) text{ and } g(x) = sumlimits_{mathbf p in mathbb{N}^d} b_{mathbf p} varphi_{mathbf p}(mathbf x)$$

I already know that the follwoing convolution formula holds :

$$(f star g)(x) = sumlimits_{mathbf p in mathbb{N}^d} c_{mathbf p} varphi_{mathbf p}(mathbf x)$$

where the coefficients are given by:

$$c_{mathbf p} = sqrt{2}^{-d}sum_{mathbf epsilon in {0,1}^d} (-1)^{lvert mathbf epsilon rvert} sumlimits_{mathbf k le mathbf p – mathbf epsilon} a_{mathbf p} b_{mathbf p – mathbf epsilon – mathbf k}$$

Suppose now that $f$ and $g$ belong to smooth laguerre balls, i.e there exists $mathbf r(f),mathbf r(g) in mathbb R_{+}^d$ and $L(f),L(g) >0$ such that :

$$sum_limits{mathbf p in mathbb N^d} a_{mathbf p}^2 e^{langle mathbf r(f),mathbf prangle} le L(f)$$

and same thing for $g$. This ensure some bound on the truncation error, which is what I am after.

Question: Can I find constants $mathbf r(fstar g)$ and $L(fstar g)$ such that the same bound applies to the convolution? Moreover for the n-convolutions of $f_1,…,f_n$ ( by recursion, or better if posible).

discrete mathematics – Use the binomial theorem and the convolution identity to express in compact form the expression for the product of

Sum of c^(2(n-k))*(2n choose 2k)x^(2k) from 2k=0 to 2n with sum of c^((2(n-k))-1)(2n choose 2k+1)*x^(2k+1) from 2k+1=0 to 2n where c is an element of the set of rational numbers not 0

I found the compact form for both sums to be:
((c+x)^2n+(c-x)^2n)/2 = Sum of c^(2(n-k))*(2n choose 2k)x^(2k) from 2k=0 to 2n
((c+x)^2n-(c-x)^2n)/2 = Sum of c^((2(n-k))-1)
(2n choose 2k+1)*x^(2k+1) from 2k+1=0 to 2n

But at this point, I’m stuck. I don’t know how to use the convolution identity here to express the product of the polynomials.

I would appreciate it if you could give me some insight as to what the right approach should be here.

Apologies for not formatting this (not familiar with the Tex methods).