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.
139
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Solving an Integral Equation Using Neumann Series in $C([0,1])$
Question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by
$$
T_K(u)(x)=\int_V K(x, y) u(y) d y
$$
Let $U=V=[0,1] \subset \mathbb{R}$. Show ...
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18
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How to Prove $T_K\left(B_1(0)\right)$ is Precompact in $C(U)$ for a Kernel Operator $K$ in $\mathbb{R}^d$
Question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by
$$
T_K(u)(x)=\int_V K(x, y) u(y) d y .
$$
Prove that $T\left(B_1(0)\right)$ is ...
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34
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How to Prove Linearity and Compute the Norm of $T_K$ for a Compact Kernel in $\mathbb{R}^d$?
Hi all this is my question: Let $U, V \subset \mathbb{R}^d$ be compact, $K \in C(U \times V)$ and $T_K: C(V) \rightarrow C(U)$ given by
$$
T_K(u)(x)=\int_V K(x, y) u(y) d y .
$$
(i) Show that $T_K \in ...
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18
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Available methods to prove that a specific integral operator has a countable point-spectrum
I'm trying to gather general techniques that may be used to prove that a specific linear operator has a countable point spectrum.
For some operators, generic theorems apply. For instance compactness, ...
- 75
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28
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Compactness of integral operators
came across this exercise I haven't been able to solve. I saw a very similar exercise (to prove an integral operator with a kernel is compact), but it is just different enough, I think, to warrant its ...
- 445
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Is the point-spectrum of bounded linear integral operators always countable?
I would like to know whether bounded linear integral operators (defined on a separable Hilbert space of functions) always have a countable point-spectrum. Or if not, what would be a practical ...
- 75
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General properties of integral operators
I study integral operators and I have several questions for which I didn't find answers.
I'm looking for references in the first place, books, where this questions discussed.
Here are the questions:
...
- 412
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28
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Conditions for an integral operator to be a compact operator on L^2(R)
I think Conditions for an integral operator to be a compact operator on L^2(R). I know there are some conditions such as Hilbert-Schmidt integral operator is compact. However, do there exist other ...
- 33
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Bounded transition kernel and finite transition kernel
I have been studying transition kernels recently from "Probability and Stochastics" by Erhan Cinlar. To avoid any ambiguity, a transition kernel in my textbook is defined as a mapping
\begin{...
- 311
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Operator in function spaces
Suppose I define the integral Operator $T:L^{\phi}(X)\to L^{\psi}(X)$ by $(Tf)(x)=\int_{X}K(x,y)f(y)d\mu$
- 75
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Integral representability of compact operators on L_2
Let $\Omega$ be a bounded Lipschitz domain and $K: L_2(\Omega) \to L_2(\Omega)$ a compact linear operator. Does $K$ in general have an integral representation, i.e., is there an integrable function $k ...
- 45
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Boundedness of integral operator induced by kernel $K(x,y) := \frac{1}{x+y}$.
Let $t_0 > 0$, $p \in (1,+\infty)$ and define $K:(0,t_0)\times (0,t_0) \rightarrow \mathbb{R}$ by $K(x,y) := \frac{1}{x+y}$. Is it true that for all $f \in L^p\bigl((0,t_0);\mathbb{R}\bigr)$ the ...
- 31
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Equivalence between trace-class and convergence of the kernel integral.
It is known that when a bounded integral operator $T$ on $L^2(S)$ with kernel $k: S^2\rightarrow \mathbb C$ is trace-class, then we have (See for instance Functional Analysis, Peter Lax, p346) $$\mbox{...
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What happens to a kernel matrix if you divide each row by its sum?
Assume we have a kernel function k(x,y), and calculate the kernel matrix
$ K = K_{ij} = k(x_i,x_j)$ for a finite dataset consisting of m points. One interest of mine is to calculate the eigenvalues ...
- 13
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73
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Find adjoint to integral operator from $H^1$ to $L_2$
Let $k(x, y): \mathbb{R}^2 \to \mathbb{R}$ be a kernel and $T: H^1(a, b) \to L_2(c, d)$
$$ T u(x) = \int\limits_{a}^{b} k(x, s) u(s) ds$$
Find the adjoint operator $T^*$.
It is easy to see the if $T: ...
- 41
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answers
71
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Spectrum of the integral operator $A(f)(x)=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$ where $A:L^2(0,2\pi)\to L^2(0,2\pi)$
I want to understand the spectrum of the integral operator $A$ from $L^2(0,2\pi)$ to itself, given by $$A(f)(x):=\int_{[0,2\pi]}\frac{\sin(n\frac{x-y}{2})}{\sin(\frac{x-y}{2})}f(y)dy$$ where $n$ is a ...
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Finding norm of the integral operator with non-integral part
Studying functional analysis, I was asked if I could determine the norm of the following functional $l:C[0,\pi]\rightarrow \mathbb R$
$$l(x)=x(0)-x(\pi/4)+\int\limits_{0}^\pi x(s)\sin sds$$
I'm ...
- 412
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163
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Compactness of the convolution integral operator
I have a problem in an exercise and I would be grateful for hints. I have to show that if $ f \in L^1(\mathbb{R}^n)$, the integral operator $$\begin{array}{rcl} T_f: L^p(\Omega)& \to& L^p(\...
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2
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225
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Integral operator and the kernel function
Let $D$ be an open bounded set in $\mathbb{R}^{n}$. Let $p \in [1,\infty)$, and $q$ is the dual exponent to $p$. Assume that $K : \mathbb{R} \times D \rightarrow \mathbb{R}$ is a bounded continuous ...
- 23
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91
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Bounding the trace-class norm of integral operator on $L^2(\mathbb{R})$
Consider an integral operator on $L^2(\mathbb{R})$ with kernel $K : \mathbb{R}^2 \to \mathbb{C}$. My question is if it is possible to bound the trace-class norm of $T$ based on properties of $K$?
This ...
- 133
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71
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Krein-Rutman theorem for kernel operators
For a matrix $A \in \mathbb{R}^{n \times n}$, we have the well-known Perron-Frobenius-Theorem
which among other things establishes the following properties:
If $A$ is positive (i.e. $A_{ij}>0 \ \...
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27
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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 ...
- 19
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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 ...
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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
$$TX=\int_0^1e^{(1-s)H}Xe^{sH}\mathrm{d}s,\qquad X\in\mathbb{R}^{d\times d}_{\...
- 411
1
vote
2
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105
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Kernel operators on AL-spaces
Let $E$ be an AL-space. For simplicity $E=L_{1}(X,\Sigma,\mu)$, where $(X,\Sigma,\mu)$ is a strictly localizable measure space. Let $T:E\rightarrow E$ be a bounded kernel operator on $E$ with ...
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356
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Compact integral operator on $H^1(\mathbb{R})$
Consider the operator
$$
{\mathcal{L}}v=e^{-x}\int_{0}^x v(y)\, dy.
$$
Is the operator ${\mathcal{L}}$ compact as an operator from $H^1({\mathbb{R}^+})$ to itself?
To give some context to the problem ...
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108
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Structure of the Inverse of a Fredholm integral operator of the second kind
NOTE: Cross-posted on MathOverflow
I am trying to solve an equation of the form
$$ (\mathbb{I} + K)\phi = f $$
where $(\mathbb{I} + K): L^2([0,1];\mathbb{R}) \rightarrow L^2([0,1];\mathbb{R}) $ is a ...
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integral operator with continuous integrable function
Let $k: [0,\infty) \mapsto \mathbb{R}$ be continuous and
$\int_{0}^{\infty} |k(x)|dx < + \infty$.
Set
$(Af)(x)=\int_{0}^{\infty} k(x+y) f(y) dy, f \in L^2(\mathbb{R})$.
Show that $A$ is bounded and ...
- 245
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1
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Examples of Continuous Functionals and Operators
I am currently working trough the chapter 2.10 in the book "Spectral Theory of Self-Adjoint Operators in Hilbert Space" (https://link.springer.com/book/10.1007%2F978-94-009-4586-9).
This ...
0
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1
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183
views
Compactness of convolution operator
Let $k$ be a bounded continuous function that is strictly positive and $\lim_{|x|\to\infty} k(x) \to 0$ and also $k \in L^1(\mathbb R)$. The question is, if the integral operator $T: C_b^0(\mathbb R) \...
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Is the integral operator with continuous kernel from $H^{-\frac 1 2}(\partial\Omega)$ to $H^{\frac 1 2}(\partial\Omega)$ a compact operator?
For a bounded domain $\Omega$ with $C^1$ boundary, define a integral operator $A$ with a continuous kernel $k$ satisfies
$$
A f (x) = \int_{\partial\Omega} k(x,y)f(y)~{\rm d}y \quad {\rm for }~x\in\...
- 11
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0
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141
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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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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) ...
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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.
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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
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1
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73
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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 ...
- 247
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139
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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}
...
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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. ...
- 22k
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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$$
Prove that $T$ is well defined.
So essentially what I want to prove is that $...
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368
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Integration is a compact operator on $L^p([0, 1])$
Let $p \in [1, \infty]$. I want to prove that this integral operator is compact:
$$
T_p: L^p([0, 1]) \to L^p([0, 1]), \quad T(f(x)) := \int_0^x f(t)dt
$$
I can prove it for $L_1$ case and I can prove ...
- 143
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45
views
Proving an expression for $T^n.$
Let $\mathcal H = L^2[0,1].$ Define an operator $T : \mathcal H \longrightarrow \mathcal H$ by $Tf(x) = \displaystyle {\int_{0}^{x} f(y)\ dy.}$ Show that for all $n \geq 1$ $$T^n f(x) = \int_{0}^{x} \...
- 2,142
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1
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142
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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 ...
user860263
0
votes
1
answer
101
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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
332
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 \...
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0
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121
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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 $[-L,L]$ ...
- 225
2
votes
0
answers
193
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
$$A\,\phi(x) = \...
- 2,930
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0
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22
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Positive definiteness wrt different borel measures
Let $K$ be a compact set in $\mathbb R^d$. Let $dx$ represent the usual Lebesgue measure and let $Q$ be a compact strictly positive definite integral operator $L^2(K,dx)\to L^2(K,dx)$:
$$Q\phi(x) = \...
- 2,930
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229
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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
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0
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134
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 ...
- 2,930
0
votes
1
answer
48
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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 $p$-...
- 1,046