Questions tagged [integral-operators]

This tag is for questions relating to integral operators, which are an important special class of linear operators that act on function spaces.

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Is the convolution with a Gaussian kernel injective?

I am a beginner in the theoretical aspects of kernels. From some tutorials I find that the Fourier transform of a Gaussian kernel is another Gaussian, which is non-zero everywhere in the frequency ...
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Sub-majorization for functions on measure spaces

I want to learn about majorization and submajorization theory on $\sigma$-finite measure spaces. I know things get a bit more complicated compared with the case of finite measure spaces, but I'm ...
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93 views

What happens to a kernel matrix if you divide each row by its sum?

Assume we have a kernel function k(x,y), and calculate the kernel matrix $K = K_{ij} = k(x_i,x_j)$ for a finite dataset consisting of m points. One interest of mine is to calculate the eigenvalues ...
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What is the correct formula of the kernel of the A$^*$A

If the kernel of the linear integral operator A on L$^2$(0,1) where A is a linear bounded integral operator non self adjoint? Thanks.
1 vote
104 views

A Schwartz kernel theorem for locally compact groups?

I am working through the paper A General Theory of Equivariant CNNs on Homogeneous Spaces. The paper is primarily aimed at a computer science audience. Here is the setting: we have a locally compact ...
• 11
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Showing that $||K^nf||_p \leq \frac{1}{n!}||k||_\infty^n ||f||_p$ where $K$ is the integral operator coming from the kernel $k(x,y) = \max\{0,x-y\}$

Here is what is written in my notes: $\bf{Remark.}$ There are quite a few ways to manufacture operators that contain only $0$ in the spectrum: Let $I = [0,1]$ and let $k \in \mathcal{C}(I^2)$ such ...
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1 vote
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Hilbert-Schmidt Integral Operator with Missing Eigenfunctions

I'm having some issues with the spectral decomposition of the integral operator $$(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with f\in L^2[0,1]}.$$ Since ...
1 vote
62 views

Pullback of integral kernel operator: how to remember the formula?

Every once in a while I have to rederive the formula for the pullback of a Euclidean integral kernel operator by a diffeomorphism. Let me focus on a simple case to make the discussion concrete. ...
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1 vote