Questions tagged [integral-operators]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
1
vote
0answers
81 views

Bifurcation points in parametric Hammerstein Integral equation

I am a theoretical physicist working on integrable systems, hence I preemptively ask forgiveness for the eventual lack of mathematical rigor. My question concerns the properties of a particular ...
1
vote
0answers
43 views

How does the constant bound on Hormander's $L^2$ estimate for non-degenerate phase depend on the cut-off function?

A classical estimate, due to Hormander, assets that the integral operator $$Tu = \int_{\mathbb{R}^{d}} e^{i\varphi(x,y)/h} a(x,y) u(x) dx, \ \ a \in \mathcal{C}_{c}(\mathbb{R}^{2d}), \ \ \varphi \in \...
0
votes
1answer
120 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....
1
vote
0answers
52 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 ...
0
votes
1answer
59 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 ...
2
votes
1answer
149 views

A numerical calculation for an integral

I am interested in the numerical calculation of $$ F(\eta)=\frac{2}{π}\int_0^{+\infty}\sin(t\eta-t^3)\frac{dt}{t}\quad\text{for $\eta\ge 0$}. $$ I believe that the function $F$ is bounded, but I do ...
0
votes
0answers
85 views

What kernel function yields power law eigenfunctions

Suppose I have a kernel function $K(x, y)$. I can then define an integral transform as follows: $$K[f] = \int_0^\infty K(x, y) f(x) dx$$. Is there any kernel function where the eigenfunctions $f(x) =...
1
vote
0answers
83 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 $...
0
votes
0answers
31 views

Integral operator on the unit disk

Assume that $a\ge 0$ and let U be the unit disk on the complex plane. Let $K(z,w) = 1/|1-z \bar w|^{2+a}$ be the kernel of integral operator T defined on the unit disk. For which $p\ge 1$and $a$ this ...
1
vote
0answers
36 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 ...
3
votes
0answers
130 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 [...
3
votes
1answer
311 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 $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
2
votes
0answers
111 views

Limit involving singular kernel: $\lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
1
vote
0answers
55 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-}^...
-1
votes
1answer
88 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(...
2
votes
2answers
163 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.
3
votes
1answer
92 views

Coupled partial differential and integro-differential equation

I have derived two equations of the following type $$ \dfrac{\partial A}{\partial x}=a\dfrac{\partial B}{\partial t}-b\dfrac{\partial^3 B}{\partial x^2 \partial t}$$ and $$ \dfrac{\partial B}{\partial ...
2
votes
0answers
64 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 $\lambda\in\mathbb R$, ...
2
votes
1answer
165 views

Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
3
votes
1answer
153 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)...
1
vote
2answers
92 views

Delta-distribution composed with a function from the Fourier representation

A well known representation of Dirac's delta-distribution is via the Fourier transform of distributions: \begin{equation} \delta[f]:=f(0)=\int_{\mathbb{R}}\int_{\mathbb{R}} e^{\mathrm{i}xk}f(k)\mathrm{...
2
votes
0answers
43 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 ...
1
vote
0answers
69 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}...
2
votes
0answers
40 views

A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: $$ f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0 $$ My question is : under what condition ...
8
votes
1answer
237 views

Cauchy path integral as a linear operator: kernel and image?

Let $\mathcal O(\Omega)$ be the algebra of functions holomorphic on the open set $\Omega\subset\mathbb C$. For $\gamma$ a simple compact curve in $\mathbb C$ consider the linear operator given by path-...
5
votes
1answer
287 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(\...
0
votes
1answer
47 views

Does $K^{1/2} (t,s)$ inherit the continuity of $K(t,s)$?

Assume that $K(t,s)$ is a (1) symmetric, (2) continuous, and (3) positive definite kernel on $[0,1] \times [0,1]$. The spectral decomposition of $K(t,s)$ is: $$ K (t,s) = \sum_{i=1}^\infty \lambda_i \...
3
votes
0answers
107 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 $\mathbb{R}^d$....
3
votes
1answer
139 views

Which utility functions are linearly transformed by normal perturbations?

I'll ask this question as pure economics, as pure math, and showing the translation. Economics (micro): Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...
1
vote
0answers
85 views

Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein ...
6
votes
1answer
258 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 ...
1
vote
0answers
67 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 ...
2
votes
0answers
76 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
vote
3answers
237 views

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

I asked the following question on math.SE (https://math.stackexchange.com/questions/2420298/bvps-for-elliptic-pdos-when-do-green-functions-l2-inverses-define-pseudo-d) just over two months ago, and it ...
4
votes
1answer
273 views

Calderon-Zygmund theorem for the kernel of spherical harmonics

I don't want to write precisely the formulation of the Calderon-Zygmund theorem for singular integrals. The details are not so important here. So I consider the operator $T$ given by the following ...
1
vote
0answers
129 views

Interpretation of Smoothing Operators as $\Psi$DO's

In most of the computations I've seen involving pseudodifferential operators (referred to as $\Psi$DO's from now on) we do not have true equality. Instead we often only have equivalence of operators, ...
4
votes
0answers
106 views

Asymptotic form of the eigenvalues / eigenfunctions of a Fredholm equation

Consider the kernel: $$k(x,y,\xi) = \sqrt{\frac{1}{\pi} \frac{1}{y+a}} \exp \left(-\frac{\xi^2}{y+a }\right)$$ I am trying to find the asymptotic form of the solutions to the following homogeneous ...
2
votes
0answers
76 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}) ...
0
votes
1answer
158 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 ...
3
votes
0answers
82 views

Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE $$\frac{\mathrm{d}^2 y}{\...
5
votes
3answers
242 views

No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?

Does there exist a continuous bilinear form $\mathcal{B}$ on $\mathcal{S}(\mathbb{R})\times \mathcal{S}(\mathbb{R})$ such that \begin{equation} \mathcal{B}(\varphi_1, \varphi_2) =\int_{\mathbb{R}\...
1
vote
0answers
53 views

Reference request: Explicit solution of Fredholm integral equations of the second kind

I am looking for explicit solutions of the following Fredholm integral equation of the second kind $$ \phi(t) = 1 + \int_0^1 G(|t-s|) \phi(s) ds, \qquad t \in [0,1], $$ for specific kernels $G$, e.g. ...
5
votes
1answer
418 views

expectation of an exponential function over a unit sphere

Let $v=(v_1,\dotsc,v_n)$ be a vector with length in $\mathbb{R}^{n-1}$, uniformly distributed over a unit sphere. I want to show $E[\exp(\alpha_n v_1)] \sim \exp (\alpha_n^2/n)$ as $n->\infty.$ If ...
0
votes
1answer
494 views

Fourier transform with cubic exponential

Please give references for the integral transform of the next kind: $$ F_3(f(x))(t)=\int_{-\infty}^{\infty} \exp(Q_3(x,t)) f(t)\,dt , $$ with $Q_3(x,t)$ - a cubic polynomial of its arguments. Special ...
1
vote
0answers
57 views

Smoothing property of integral operators

Consider an integral operator $J$ with kernel $k(x,y)$ (assuming its properties are nice), can we describe the operator $J$ in $$Jf(x) := \int_0^1 k(x,y) f(y) dy$$ as an isomorphism(i.e. $J$ has a ...
1
vote
1answer
204 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 $\mathbb{R}^n$. ...
2
votes
1answer
154 views

Regularity of solutions to certain integral equation

The so-called Symm's integral equation on an interval $[a,b]$ is defined by $$\int_a^bu(y)\log|x-y|dy=f(x),\,\,x\in[a,b],$$ and $f$ is a given function. In the introduction of a paper by I. H. Sloan ...
0
votes
0answers
54 views

$L^{\infty}$ norm of Integral-Differntial equation's solution

Let $\phi(t,z)\in C^{1,2}$ is a function taking values in $\mathbb{R}^D$ , $\rho_{i,j}$ and $\mu_i$ be vector-valued functions and consider the non-linear PDE $$ \partial_t\Phi(t,z) - \phi(0,z) - \...
7
votes
2answers
358 views

Infinite dimensional version of a simple fact on certain singular matrices

We consider the following simple fact about matrices. Then we try to generalize it in the context of smooth manifolds; Let $L$ be the collection of all $n \times n$ real matrices $A=(a_{ij})$ with ...
1
vote
1answer
90 views

Well-definedness for a singular integral

Let $T_\alpha$ be a singular integral operator defined by $$ T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds $$ for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$. ...