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.
125
questions
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11
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Eigen functions of a non self-adjoint integral operator.
Let us define the operator from the set of smooth complex valued functions from $[0,1]$ to itself as
$$L[f](u) = \int_{0}^{1} e^{-j2\pi(y-x)^2} f(x)dx, \quad 0\leq u \leq 1 .$$
Here $j=\sqrt{-1}$.
I ...
- 713
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votes
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answers
18
views
Asymptotics of singular values of the operator
In space $L^2(B_R(0))$ consider operator $Tx(t)=\displaystyle\phi(t)\int\limits_{B_R(0)}\cos(|t|^{1/2}|s|^{1/6})\psi(s)x(s)ds$, where $\phi, \psi \in L^2$. It is clear that this operator is compact. ...
- 492
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38
views
If-and-only-if condition on the kernel for an integral operator $T:L^2 \rightarrow L^2$ to be compact
Let $\Omega \neq \emptyset$ and $\mu$ be a finite measure on $\Omega.$ We are cosidering a kernel $k$ on $\Omega,$ i.e. a symmetric, non-negative definite jointly measurable function
$$k: \Omega \...
- 68
0
votes
0
answers
18
views
An eigenfunction inequality for integral operators
I have an integral operator $T$ defined with respect to a positive semidefinite kernel function $k: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ probability measure $\mu(dx) = p(x) dx$ defined ...
- 21
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votes
1
answer
13
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How to prove that the range of this integral operator $T$ is $L^2[a,b]$?
The operator is $Tf= g(t)+ \int_{a}^{t}(K(t,s)f(s)ds.$
The proof assumes that $g(t) \in L^2[a,b]$ so I believe I need to only prove that $\int_{a}^{t}(K(t,s)f(s)ds \in L^2[a,b]$.
I started off this ...
- 128
1
vote
0
answers
36
views
Averaged matrix exponential inverse
Let $H$ be a given symmetric matrix, i.e., $H\in\mathbb{R}^{d\times d}_{\rm sym}$, $d\geq1$. The operator $T$ is given by
$$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
- 356
1
vote
2
answers
63
views
Kernel operators on AL-spaces
Let $E$ be an AL-space. For simplicity $E=L_{1}(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a strictly localizable measure space. Let $T:E\rightarrow E$ be a bounded kernel operator on $E$ with ...
- 31
6
votes
1
answer
203
views
Compact integral operator on $H^1(\mathbb{R})$
Consider the operator
$$
{\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy.
$$
Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself?
To give some context to the problem ...
- 1,008
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0
answers
37
views
Calculating Trace of Integral Operators
I could not figure out how the following formula can be derived:
$$ \operatorname{Tr} [ W(t) Q(t) ] = \frac{1}{(2\pi)^{d/2}} \int_{\mathbb{R}^d \times \mathbb{R}^d} \widehat{W}(t, q-p) \widehat{Q}(t,p,...
- 102
2
votes
0
answers
50
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Structure of the Inverse of a Fredholm integral operator of the second kind
NOTE: Cross-posted on MathOverflow
I am trying to solve an equation of the form
$$ (\mathbb{I} + K)\phi = f $$
where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\mathbb{R}) $ is a ...
- 21
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votes
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38
views
Compactness integral operator
The operator
$Af(x)=\int_{\sqrt{x}}^{1} \frac{1}{x+y}f(y) dy, \quad f \in L^{2}([0,1])$,
is bounded on $L^2([0,1])$, with
$||Tf||^2=\int_{0}^{1} |\int_{\sqrt{x}}^{1} \frac{1}{x+y}f(y) dy|^2 dx \leq ......
- 272
2
votes
1
answer
58
views
integral operator with continuous integrable function
Let $k: [0,\infty) \mapsto \mathbb{R}$ be continuous and
$\int_{0}^{\infty} |k(x)|dx < + \infty$.
Set
$(Af)(x)=\int_{0}^{\infty} k(x+y) f(y) dy, f \in L^2(\mathbb{R})$.
Show that $A$ is bounded and ...
- 272
1
vote
2
answers
33
views
Examples of Continuous Functionals and Operators
I am currently working trough the chapter 2.10 in the book "Spectral Theory of Self-Adjoint Operators in Hilbert Space" (https://link.springer.com/book/10.1007%2F978-94-009-4586-9).
This ...
0
votes
1
answer
56
views
Compactness of convolution operator
Let $k$ be a bounded continuous function that is strictly positive and $\lim_{|x|\to\infty} k(x) \to 0$ and also $k \in L^1(\mathbb R)$. The question is, if the integral operator $T: C_b^0(\mathbb R) \...
- 3
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votes
0
answers
14
views
How Can One Express u(xy) as a Diagonalized Transform Kernel, K(x,y)?
Consider a projection operator $P_{u}g(x)=<g(x),u(x)>$, where $u(x)$ is an eigenfunction normalized under an inner product, $<u_{m}(x),u_{n}(x)>=\delta_{m,n}$.
(ASIDE: Inner products may ...
- 39
1
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0
answers
40
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Is the integral operator with continuous kernel from $H^{-\frac 1 2}(\partial\Omega)$ to $H^{\frac 1 2}(\partial\Omega)$ a compact operator?
For a bounded domain $\Omega$ with $C^1$ boundary, define a integral operator $A$ with a continuous kernel $k$ satisfies
$$
A f (x) = \int_{\partial\Omega} k(x,y)f(y)~{\rm d}y \quad {\rm for }~x\in\...
- 11
1
vote
0
answers
95
views
Numerical methods for computing singular value decomposition of integral operator
Consider a Hilbert-Schmidt integral operator $T:L^2(\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n)$, whose explicit mapping is given by
\begin{align}
f(y)\mapsto \int k(x,y) f(y) dy.
\end{align}
Hopefully,...
- 339
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answers
31
views
Lower-bound for $\inf_{f \in H} \|\nabla f\|_{L^2(\tau_d)}/\|f\|_H$ for an RKHS $H$ induced by dot-product kernel
Let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $\tau_d$ be the uniform probability measure thereupon. Let $H_K \subseteq L^2(\tau_d)$ be an RKHS of square $\tau_d$-integrable functions, ...
- 8,419
2
votes
0
answers
37
views
Kernel of Continuous Functional Calculus of Integral Operator
Suppose we have a symmetric bounded function $k:[0,1] \times [0,1] \rightarrow \mathbb{R}$ that induces the integral operator $T_k: L^2([0,1]) \rightarrow L^2([0,1])$,
$$
(T_k f)(u) = \int_0^1 k(u,v) ...
- 167
0
votes
1
answer
93
views
What is the correct formula of the kernel of the A$^*$A
If the kernel of the linear integral operator A on L$^2$(0,1) where A is a linear bounded integral operator non self adjoint?
Thanks.
- 41
1
vote
0
answers
47
views
A Schwartz kernel theorem for locally compact groups?
I am working through the paper A General Theory of Equivariant CNNs on
Homogeneous Spaces. The paper is primarily aimed at a computer science audience.
Here is the setting: we have a locally compact ...
- 11
0
votes
1
answer
30
views
Showing that $||K^nf||_p \leq \frac{1}{n!}||k||_\infty^n ||f||_p$ where $K$ is the integral operator coming from the kernel $k(x,y) = \max\{0,x-y\}$
Here is what is written in my notes:
$\bf{Remark.}$ There are quite a few ways to manufacture operators that contain only $0$ in the spectrum: Let $I = [0,1]$ and let $k \in \mathcal{C}(I^2)$ such ...
- 217
1
vote
0
answers
83
views
Hilbert-Schmidt Integral Operator with Missing Eigenfunctions
I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
...
- 23
1
vote
1
answer
35
views
Pullback of integral kernel operator: how to remember the formula?
Every once in a while I have to rederive the formula for the pullback of a Euclidean integral kernel operator by a diffeomorphism. Let me focus on a simple case to make the discussion concrete. ...
- 20.9k
1
vote
1
answer
51
views
Prove the operator $\int_0^x (x-t)f(t)dt$ is well defined.
Consider the operator $T : C[0,1] \to C[0,1] $ ($C[0,1]$ has the maximum metric)
defined as $$T(f)=\int_0^x (x-t)f(t)dt$$
Prove that $T$ is well defined.
So essentially what I want to prove is that $...
- 383
4
votes
1
answer
220
views
Integration is a compact operator on $L^p([0, 1])$
Let $p \in [1, \infty]$. I want to prove that this integral operator is compact:
$$
T_p: L^p([0, 1]) \to L^p([0, 1]), \quad T(f(x)) := \int_0^x f(t)dt
$$
I can prove it for $L_1$ case and I can prove ...
- 123
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votes
1
answer
42
views
Proving an expression for $T^n.$
Let $\mathcal H = L^2[0,1].$ Define an operator $T : \mathcal H \longrightarrow \mathcal H$ by $Tf(x) = \displaystyle {\int_{0}^{x} f(y)\ dy.}$ Show that for all $n \geq 1$ $$T^n f(x) = \int_{0}^{x} \...
- 1,751
0
votes
1
answer
78
views
Kernel of a linear functional
I have $L^2((0,1))$ space and functional
$$
F(f) = \int_0^1 f(x)(1-x) dx.
$$
How do I find the kernel?
From definition we want to find $f$ such that $\int_0^1 f(x)(1-x) dx = 0$. I suspect we want the ...
- 187
0
votes
1
answer
38
views
How to prove that eigenfunction of translationally invariant continuous operator $ K(t-t') $ is $ \exp(iwt) $?
I was studying a book in computational neuroscience where I came across the following equations:
$$ \int{W(t,t')e(t')dt'}=\lambda e(t) $$ and read that if $ W(t,t')=K(t-t') $ then the eigenfunction $ ...
1
vote
1
answer
183
views
Compute the spectrum of an explicit integral operator
Let $H = L^2[0,1]$. Define an operator $K \in B(H)$ by $Kf(x) := x^2 \cdot \int_{0}^{1} y f(y) \; \text{d} y$. Show that $K$ is compact and compute its spectrum.
I already showed that $ ||K || \leq \...
- 29
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votes
0
answers
67
views
Dirac $\delta$-function as a limit of matrices, and eigenvalues of integral kernels.
I am interested in calculating the eigenvalues of integral kernels, but I figured it would be useful to get some intuition about the simplest one first: the identity operator.
The interval $[-L,L]$ ...
- 215
2
votes
0
answers
113
views
Mercer's theorem for zonal kernel
Let $f$ be a continuous function $[-1,1]\to\mathbb{R}$. Consider an integral operator $A$ on the unit sphere $S^{d-1}$ of $\mathbb{R}^d$, which acts on $\phi\in\mathcal{L}^2(S^d)$ as
$$A\,\phi(x) = \...
- 2,820
1
vote
0
answers
13
views
Positive definiteness wrt different borel measures
Let $K$ be a compact set in $\mathbb R^d$. Let $dx$ represent the usual Lebesgue measure and let $Q$ be a compact strictly positive definite integral operator $L^2(K,dx)\to L^2(K,dx)$:
$$Q\phi(x) = \...
- 2,820
0
votes
1
answer
95
views
Interchange Integration and Minimization
I am new to calculus of variations and I have a problem of the type
\begin{equation*}
\text{min}_{g\in L^2(\mathbb{R})} \int_{\mathbb{R}} f(x,g(x))\, \mathrm dx,
\end{equation*}
where $f$ is ...
- 15
1
vote
0
answers
61
views
Mercer theorem and uniformly bounded continuous function
Consider a continuous symmetric real function $k$ on $I\times I$, where $I$ is a compact real interval. Let $K$ be the integral operator whose kernel is $k$. Assume that $K$ is strictly positive ...
- 2,820
0
votes
1
answer
38
views
Example of a "weak" Compact operator
Let $\mathcal{H}$ be a separable Hilbert space with ONB $\{e_n\}_n$. Following my previous question, I would like to find an example of a bounded operator $A$ such that
$A$ is compact.
$A$ is not $p$-...
- 924
2
votes
1
answer
99
views
Extension of the Schur product theorem to operators
Given two $n\times n$ matrices $A$ and $B$, define their Hadamard product $A\circ B$ as the element-wise product, i.e.
$$(A\circ B)_{ij} = A_{ij}B_{ij}\,.$$
A well known result is the Schur product ...
- 2,820
2
votes
1
answer
115
views
Conditions for compactness of operator
Let $A$ be an bounded operator on a Hilbert space with ONB $\{e_n\}_n$. I am looking for precise conditions on $\langle e_n, A e_m \rangle$ to guarantee that $A$ is compact (i.e. the limit of finite ...
- 924
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votes
0
answers
41
views
Integral calculation for eigenvalues
I am trying to solve the following integral
$\int_{0}^{2\pi}{\left|\int_{0}^{2\pi}{e^{ik\cos\left(\theta-\varphi\right)}\cos\left(n\theta\right)}d\theta\right|^2 d\varphi},\ \ n=0,1,2,.......$
which ...
1
vote
1
answer
72
views
Integral operator is not closable
Let $A$ be a linear operator from $X$ to $Y$ with domain $D(A)$. I've learned the following characterization for closable operators: $A$ is closable if and only if for every sequence $(x_n) \in D(A)$ ...
- 61
2
votes
0
answers
243
views
Find spectrum of integral operator in L2
Please help me find the spectrum of operator $A$, I do not speak English well. In Russian forums I did not find the answer.
Find spectrum of integral operator in $L_2$($R$):
$$ (Ax)(t) = \int\limits_{-...
- 21
0
votes
1
answer
107
views
Prove that a specific non self-adjoint linear operator has real eigenvalues
I am dealing with a Fredholm operator $A$, whose kernel $K$ is non symmetrical. However when I discretise A and compute the eigenvalues of the non symmetric matrix I obtain, I observe that A only has ...
- 45
0
votes
1
answer
87
views
Contraction mapping in $C([0,1])$
Suppose $T$ is an operator on $C([0,1])$ defined by $(Tu)(t) = \displaystyle\int_{0}^{t} u(x)^2\,\mathrm dx$. Show that $T$ is a contraction mapping on the closed ball of radius $\dfrac14$ in $C([0,1])...
3
votes
3
answers
876
views
How to Find the Spectrum of an Integral Operator
I need to find the spectrum of an operator $T: C([0,1]) \to C([0,1])$ defined by $(Tf)(t) = \int_0^t f(x) dx$. I know that the spectrum is the set of all values $\lambda$ such that $\lambda I - T$ is ...
- 177
2
votes
1
answer
63
views
Are Hilbert Scmidt integral operators on separable compact Hausdorff spaces in the Hilbert Schmidt class?
Let $X$ be compact separable Hausdorff space with a positive Borel measure $\mu$. Assume $L^2(X)$ is separable.
Consider a function $K: X \times X \to \mathbb{C}$ with $K \in L^2(X \times X , \mu \...
- 35
1
vote
2
answers
399
views
Integral operator has no eigenvalue
Let $V$ be the vector space of all real valued continuous functions. Prove that the linear operator $\displaystyle\int_{0}^{x}f(t)dt$ has no eigenvalues.
This question is same as
Prove that the ...
- 29
7
votes
1
answer
205
views
An Orthogonality Problem of Eigenfunctions of homogeneous Fredholm equation
Suppose we have a integral equation $$\int_{-1}^1 \frac{\text{sin }c(x-y)}{\pi (x-y)}\psi(y)dy=\lambda \psi(x),\quad|x|\le1.$$
By the Fredholm equation theory, we know that this equation has ...
- 1,491
1
vote
1
answer
250
views
Eigenvalues and eigenfunctions of an integral operator
Let $T$ be an integral operator with kernel $K(x,y)=e^{|x-y|}$ on $L^2(-1,1)$. How can we find the eigenfunctions and eigenvalues of $T$?
Even though I am not sure whether the following arguments are ...
- 2,645
4
votes
1
answer
139
views
Point spectrum of an integral operator
Let we have $$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy$$
so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
- 2,645
0
votes
1
answer
323
views
Operator norm of integral operator
Suppose we have $X=L^2([0,1];\mathbb{R})$ and
\begin{equation}
T:X\rightarrow X, \ Tf(x)=\int_0^1x^2yf(y)dy.
\end{equation}
Show that $T$ is compact and determine $||T||.$
I already have that $||T|...
- 107