All Questions
Tagged with fa.functional-analysis integral-operators
59 questions
2
votes
0
answers
52
views
On distributions and kernels
Let $U\subset\mathbb{R}^{d}$ be an open set and consider $X=\mathbb{R}\times U$. Now, lets consider a smooth (regular) kernel $k_{A}\in C^{\infty}(X\times X)$ and corresponding continuous operator $A:...
- 1,429
1
vote
1
answer
117
views
Lower bound for a commutator trace
I have this Hilbert space of square-integrable complex-valued functions on a square, $\mathbb{L}^2([0,1]^2)$. And let $M_x$, $M_y$, and $M_{x+y} = M_x+M_y$ be the operators of multiplication by the ...
- 121
0
votes
0
answers
121
views
How to find the inverse of this linear integral operator?
Let $f(x): \mathbb{R}^d \rightarrow \mathbb{R}$ be a function that decays ``fastly enough'' at infinity.
We can define the following linear operator
$$L[f](x):= \int_{\mathbb{R}^d} d^d y \, \frac{f(y)}...
- 1
1
vote
1
answer
130
views
Existence of solutions to a series of integral equations
I am trying to solve the following integral equation analytically:
$$
\sum_{n \geq 1} \left( \int_0^te^{-n^2(t-s)} f_n(s) \, ds \right) = g(t), \quad t \in [0, T],
$$
where $(f_n(t))_n$ is the unknown ...
- 617
1
vote
0
answers
45
views
Characterization of the Picard's condition for integral equation
Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
- 143
2
votes
0
answers
73
views
Derivative of a functional involving integral and level set
Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional
$$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$
where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
- 29
3
votes
1
answer
293
views
How to find the inverse of a product of two integral equations
Problem
I am trying to invert an equation of the form:
$R(l_0)=(\int_{0}^{l_0} \rho(x) \, dx)(\int_{l_0}^{l} \rho(x) \, dx)$
where $0\leq l_0 \leq l$
I.e. I want to find $\rho(x)$ given $R(l_0)$ via ...
- 33
0
votes
0
answers
145
views
Why is this function in $L^1$?
I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
- 101
3
votes
0
answers
151
views
Reference request: trace norm estimate
In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$
then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
- 101
0
votes
1
answer
135
views
When integrating by part produces a singularity
I'm currently interesting in the following operator:
$$O[f](x):=f(x)-2xe^{x^2}\int_x^{+\infty}dt \, e^{-t^2} f(t)$$ for all $x\in\mathbb{R}$ and $f$ smooth and decaying at infinity fast enough with ...
- 41
3
votes
0
answers
143
views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation
$$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$
Here $T(\cdot)$ is a ...
- 299
2
votes
0
answers
61
views
Translation request: Boundedness of Cauchy integral on Lipschitz boundary
The reference: "L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes" (https://annals.math.princeton.edu/1982/116-2/p04) is written in French. Can we ...
- 269
4
votes
1
answer
405
views
Why is it difficult to define a direct integral of Banach spaces or Banach algebras?
In the relevant Wikipedia entry, I can read about how to define a direct integral on Hilbert spaces and Von-Neumann algebras.
Suppose that I want to define a direct integral on either Banach spaces or ...
- 1,054
1
vote
0
answers
74
views
Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$
Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
- 63
2
votes
1
answer
585
views
Structure of the inverse of a Fredholm integral operator of the second kind
NOTE: Cross-posted on Mathematics Stack Exchange
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];\...
- 21
3
votes
0
answers
214
views
Extended adjoint of Volterra operator
Let $V$ be a Volterra operator on $L^2 [0,1]$.
Does there exist a nonzero operator $X $ satisfying the following system
$VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator?
$$ V(f) (x) =\...
3
votes
1
answer
497
views
Hilbert-Schmidt integral operator with missing eigenfunctions
I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
...
- 33
3
votes
0
answers
121
views
Schatten norm estimate of spatially truncated resolvent of Laplacian
Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form
$$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$
where $1_{\Gamma_m}$ denotes multiplication ...
- 91
2
votes
0
answers
141
views
Optimization of functionals with constraints
I have a minimization problem as follows:
$\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $
$\texttt{s.t.}\;\;\;...
1
vote
1
answer
181
views
Unique solution for 2$\times$2 Fredholm integral equations system
Consider the following system of Fredholm integral equations with constant kernel matrix
$$
f(x)=K(x)\int_{0}^{1}f(s)ds
$$
where $K(x)\in C([0,1];M_{2\times 2}(%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\...
- 617
2
votes
0
answers
126
views
Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions
I have a system of nonlinear Volterra integral equations of form
$$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$
and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
- 123
0
votes
1
answer
323
views
Solution set of integral equation/ Kernel of linear operator
I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int_0^1 \int_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in ...
- 143
2
votes
1
answer
756
views
General strategy for studying the decay of eigenvalues of kernel integral operators
Disclaimer. Please, be patient, I'm here to learn functional analysis...
Let $X$ be the unit sphere in $\mathbb R^n$ and let $\sigma$ be the uniform measure on $X$. Consider a positive definite ...
- 6,853
2
votes
2
answers
860
views
Conditions for continuity of an integral functional
Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F_f$, defined ...
- 5,405
2
votes
1
answer
719
views
Injectivity of an integral operator
Consider the operator
$$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$
with $k\in L^2((0,1)\times(0,1)).$
I want to know under what assumption the kernel is reduced to zero. i....
- 617
1
vote
0
answers
179
views
Polar decomposition of the Volterra integral operator
Repost of this Math.SE question due to a lack of answers (No one was able to help me find the closed form of $U_T$ and $|T|$ after two bounties). I also searched extensively online but couldn't find ...
- 67
0
votes
1
answer
343
views
Eigenvalues of an integral operator
Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by
$$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$
What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
- 617
3
votes
0
answers
648
views
When the square root of integral operator becomes also integral operator (with continuous kernel)?
Let $X$ be a compact metric space and $\mu$ be strictly positive Borel measure on $X$. Let $T:L^2(X,\mu)\rightarrow L^2(X,\mu)$ be a self-adjoint, compact, and positive operator on the Hilbert space $...
- 469
1
vote
0
answers
159
views
Existence of continuous integral kernel
Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$).
A ...
- 31
2
votes
0
answers
165
views
Run-away Volterra operator
For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by
$$
A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
- 5,405
3
votes
1
answer
646
views
Eigenfunctions and eigenvalues of an operator defined by a certain integral
Let $k :[0,1]^2 \rightarrow \bf{R} $ be a kernel function definded by
$ k(x,y)= (1- max(x,y))^2 .$ Now, let $ L $ be a linear operator defined on $ L^2 [0,1] $ by $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
- 245
1
vote
0
answers
67
views
Angle between Fleming-Viot type 3-particle system
Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
- 284
-1
votes
1
answer
102
views
Compactness of a special kind of Integral operators
Let $(S(t))_{t>0}$ be a continuous operator from $L^2(0,1)$ to its self and Let $K$ be the operator $$\eqalign{
& K:{L^2}(0,1) \to {L^2}(0,1) \cr
& f: \to (Kf)(x) = \int\limits_0^1 {k(...
- 617
2
votes
2
answers
209
views
A Fredholm equation with a particular kernel
How to solve this kind of Fredholm’s equation?
$$
x(t)+\lambda \int\limits_{0}^{1}\! \big[ts - \min\{t,s\}\big]x(s)ds=t
$$
Thanks for any help.
- 39
2
votes
0
answers
124
views
Solving Fredholm integral equation in Lp
I have a very simple integral equation
$$
f(x) - \lambda \int_a^be^{x-y}f(y)dy=1
$$
which is Fredholm of the 2nd kind, with separable kernel. It is needed to find the values of $\lambda\in\mathbb R$, ...
4
votes
1
answer
602
views
Trace-class properties of integral operator
Let $k$ be a smooth, compactly supported function defined on $\mathbb{R}^{2}$ and let $Op(k)$ denote the integral operator on $L^{2}(\mathbb{R})$ defined as $$Op(k)f(\cdot)=\int_{\mathbb{R}}k(y,\cdot)...
- 43
2
votes
0
answers
64
views
Integral equation with kernel defined in a rectangle
Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$
Observe that the kernel is not defined on a square.
My question: Can I apply the classical ...
- 617
1
vote
0
answers
131
views
Uniqueness of solution of Volterra Integral Equation with deviating argument
In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$:
\begin{equation}...
- 259
5
votes
1
answer
709
views
What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?
Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that
$$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$
Let us consider the integral operator $\mathcal{L}$ on $L_2(\...
- 51
4
votes
0
answers
254
views
Lower semi-continuity of integration
I've found many papers characterizing the weak lower semi-continuity of
$$
\Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx,
$$
on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
- 5,405
6
votes
1
answer
898
views
Injectivity of a Fredholm operator
While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
Question : Suppose $K$ is ...
- 406
1
vote
0
answers
141
views
Characterisation of functions for which the Fourier transform commutes with a particular operator
Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
- 1,199
2
votes
0
answers
88
views
Functional equation involving integrals and exponential
Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that:
$$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$
where $\lambda$ is a complex number (with $0&...
- 1,199
2
votes
0
answers
79
views
One-dimensional integral equation uniquely solvable?
I recently met a question similar to this one and I would like to post it here, because I basically found nothing:
We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
- 329
0
votes
1
answer
268
views
Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
- 329
1
vote
1
answer
392
views
Integral kernel smooth
Assume that $Tf(x):=\int_{\mathbb{R}^n} K(x,y)f(y) dy$ is an operator such that $T \in L( H^{-k}, H^k)$ is continuous for any $k$, where $H^k$ is the $k-th$ order Sobolev space on $\mathbb{R}^n$.
...
- 11
3
votes
0
answers
82
views
Eigenvalues of approximations to product-convolution operators
Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions.
This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...
- 455
5
votes
1
answer
1k
views
Fredholm operators in $K$-theory?
Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...
- 145
2
votes
1
answer
741
views
solution uniqueness of non-linear Fredholm equations
the equation is
$F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$
where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a non-...
- 273
2
votes
1
answer
436
views
Interpolation between weighted $L^p$ spaces
Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded.
Let $T$ be the (...
- 943