Questions tagged [integral-kernel]

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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];\...
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Is there any way this property of semigroups can be satisfied?

Suppose you have the heat semigroup $(S(t))_{t>0}$, such that $$S(t)u(x) = (4\pi t)^{-n/2}\int_{\mathbb{R}^n}e^{-|x-y|^2/4t}u(y)dy.$$ The semigroup has the property that $$S(t)S(s)u(x) = S(t+s)u(x)....
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If $x \mapsto m_x$ is a Markov kernel and $K$ is a psd kernel, is the RKHS of $K':(x,x') \to E_{m_x \otimes m_{x'}}K(z,z')$ contained in that of $K$?

Let $X$ be a measurable set (e.g $X = \text{euclidean $\mathbb R^n$}$, for concreteness). Let $K:X \times X \to \mathbb R^n$ be a psd kernel on $X$, and let $m:X \to \mathcal P(X)$ be a Markov kernel ...
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Properties of a kernel convolution $K'(x,y) = \int_X\int_X K_0(x,a)K(a,b)K_0(b,y)d\mu(a)d\mu(b)$ where $K$ and $K_0$ are kernels on $(X,\mu)$

Let $(X,\mu)$ be a probability measure space and $K:X \times X \to \mathbb R$ be a (psd) kernel on $X$. Let $K_0$ be another kernel on $X$ and defined a new kernel $\widetilde K$ on $X$ by $$ \...
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Knowledge on weighted integral operators?

There are tons of books and a huge literature on the properties of the following integral operator: \begin{equation} T(f) = \int_{\mathcal{X}} K(x,\cdot)f(x)dx, \end{equation} where $K(x,z)$ is, say, ...
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Radial-energy decomposition of a velocity field in 2D: is anyone able to show that the lemma below is true (…or false)?

In the book “Vorticity and Incompressible Flow” by Majda and Bertozzi, there is the following lemma. Lemma 3.2. Any smooth incompressible vector field $v$ in $\mathbb{R}^2$ with vorticity $$\omega=\...
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1 vote
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38 views

Explicit expression for the Poisson kernel solving the Dirichlet problem for geodesic balls

Let $X$ be a Riemannian manifold with the Laplace-Beltrami operator denoted by $\mathscr L$ and we look at its geodesic balls say $B$. Let $u$ be a continuous function on the geodesic sphere which is ...
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Extending Ky Fan's eigenvalues inequality to kernel operators

--Migrating from MSE since it might fit better here-- Base result The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as: $$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
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Nonlinear variation of constants formula

Suppose that we wish to solve $x'(t)=f(x(t))+g(x(t)), \; x(0)=x_0\in X,$ where $X$ is an infinite dimensional Banach space and $f , g : X \rightarrow X $ are two nonlinear functions. Furthermore, ...
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Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
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Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

When we have a system of of $n$ linear equations represented by $$A \vec{x} = \vec{b} $$ with $\vec{x} = (x_{1}, x_{2}, \dots, x_{n})^{\intercal} $, we can solve for each component of this vector by ...
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1 answer
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Calculation of an inverse Mellin transform

Let $z \in C$ and consider the following integral equation: $$-\frac{\Gamma(a)}{\Gamma(b)}\frac{1}{ z \dfrac{\mathrm{d}}{\mathrm{d}z} {_{1}F_{1}}(b,b-a;z)}= \int_{0}^{+ \infty}x^{z-1}K(x) \mathrm{d}x$$...
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1 answer
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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 \...
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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 ...
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1 answer
77 views

Existence of integral kernel

I know the following statement ture. Let $T \in B(L^1(\mathbb{R}^d), L^\infty(\mathbb{R}^d))$, where $B(X, Y)$ denotes all bounded linear operoters from $X$ to $Y$. Then, $T$ has the integral kernel $...
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Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $\phi:[-1,1] \to \mathbb R$ be a function such that $\phi$ is $\mathcal C^\infty$ on $(-1,1)$. $\phi$ is continuous at $\pm 1$. For concreteness, and if it helps, In my specific problem I have $\...
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2 votes
1 answer
271 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 ...
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4 votes
1 answer
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Inverting convolutions over finite intervals

There are well-known techniques for inverting convolutions over the whole or half real line with Fourier and Laplace transformations, but on the face of it they can't be applied to an integral ...
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Approximate identities on the unit disk and going beyond a power series' radius of convergence

Let $\left\{ a_{n}\right\} _{n\geq0}$ be a bounded sequence of complex numbers, so that the power series $f\left(z\right)=\sum_{n=0}^{\infty}a_{n}z^{n}$ has a radius of convergence $\geq1$. ...
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3 votes
1 answer
216 views

Gradient condition implies Hörmander condition

We have tempered distribution $K$ in $\mathbb{R}^n$ which coincides with a locally integrable function in $\mathbb{R}^n\setminus \{0\}$. We call the condition $$\int_{|x|>2|y|}|K(x-y)-K(x)|dx\leq ...
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Is this a positive definite kernel?

Under which conditions on the function : \begin{array}{l|rcl} K : & \mathbb R^+ & \longrightarrow & (0, 1)\\ &t & \longmapsto & K(t) \end{array} is the symmetric ...
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1 vote
0 answers
62 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 \...
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0 votes
1 answer
358 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....
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420 views

Boundedness of integral operators on spaces of continuous functions

Consider a standard integral operator $T$ formally defined by $$ Tf(x):=\int_{K} k(x,y)f(y)dy,\qquad x\in K, $$ where $K$ is a locally compact metric measure space. It is immediate to see that the ...
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3 votes
0 answers
96 views

Generalizing the heat kernel approach

I notice a way of solving equations that goes roughly like this: Want to solve $Tu=v$, i.e. hoping to find "$T^{-1}(v)$". $T^{-1}$ might not exist "on the nose", so instead we consider $\int_0^\infty ...
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1 vote
0 answers
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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 ...
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3 votes
1 answer
443 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 $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...
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0 answers
60 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-}^...
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2 votes
0 answers
237 views

Convolution and approximate heat kernel

I have the following question: Let $A, B\in C^\infty((0,\infty)\times M\times M)$, where $M$ is a compact manifold. Define \begin{align} (A*B)(t,x,y) = \int_0^t\int_M A(t-s,x,z)B(s,z,y)dzds \end{align}...
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2 votes
0 answers
94 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 $\lambda\in\mathbb R$, ...
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4 votes
1 answer
314 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)...
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0 answers
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Looking for example of integral transformations that preserve number of zeros

Let $f:\mathbb{R} \to \mathbb{R} $ have $n<\infty$ zeros. I am looking for non-trivial examples of integral transformation \begin{align} g(x)= \int f(t) h(t,x) dt \end{align} such that $f$ and $g$...
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2 votes
0 answers
49 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 ...
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2 votes
0 answers
43 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 ...
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4 votes
1 answer
259 views

integral kernel function for the SU(N) group

It is well know that the Haar probability measure for the $U(N)$ group, given by $$ \begin{align} dX_{U(N)} & = \frac{1}{N!(2\pi)^N} \begin{vmatrix} 1 & 1 ...
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3 votes
1 answer
221 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 ...
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6 votes
1 answer
309 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 ...
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6 votes
1 answer
601 views

Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
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Existence theorems Volterra Equation of second kind on unbounded domains

The general Volterra Equation of the second kind in convolution form can be described by: $$ \phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a $$ Suppose we wish to ...
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1 vote
0 answers
65 views

Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?

In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
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0 answers
210 views

Is there any orthogonal basis set that satisfies an integral equation appearing in physics?

I am considering the following integral equation $$ f(x) = \int_{-W}^W \frac{\exp(-x y)}{\cosh(y)} g(y) dy, $$ where $-1 \le x \le 1$, $f(x)$ and $g(y)$ are real functions. This integral equation ...
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3 votes
1 answer
475 views

When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)

Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator $$ Lu:=-\Delta u+V(x)u $$ where $V\geq0$ is the electric potential, ...
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0 answers
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basis representation of a special sinc kernel

I have a functional map from $(x,y) \in \mathbb{R}^2$ to another function $f_{x,y}(z,w) \in \mathbb{C}$. (Variables $z,w $ range from $-\infty$ to $\infty$.) That is, for any pair $x,y$, I get a ...
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3 votes
3 answers
462 views

Gaussian bounds on Dirichlet heat kernel

Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0 $ such that $$\frac{c}{t^{n/2}} e^{-\frac{1}{4t}d(x, y)^2} \leq ...
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1 vote
1 answer
95 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 $0<\alpha<1$. ...
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2 votes
0 answers
235 views

Jacobi's Elliptic functions - Kernel

I have an integral equation with a kernel expressed in terms of Jacobi's elliptic functions. In particular I want to solve the following equation: $$\lambda \begin{pmatrix} X_1(u) \\ X_2(u) \end{...
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3 votes
0 answers
165 views

Karhunen-Loeve expansion convergence rate for Gaussian Proccess

Consider A Gaussian Procces $X(t):\mathbb{R}\times \Omega \to \mathbb{R}$ with $\Omega$ a probability space and $\mathbb{E} \left[ X_t \right] = 0$ for all $t\in \mathbb{R}$. Consider also its KL ...
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1 vote
1 answer
205 views

Is there a way to solve this integral equation?

I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem. For $\xi = (\alpha\theta)^{1/\alpha}$ and for ...
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1 vote
1 answer
247 views

Various limits of the Christoffel Darboux Kernel

In a different thread, we stumbled upon the following question: Given a continuous finite measure $w(x)dx$ on some interval $(a,b)$, $-\infty \leq a <b \leq \infty$, and the set of respective ...
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  • 3,200
5 votes
2 answers
295 views

Error estimate in the spectral theorem of compact operators on a Hilbert space

Given a compact self-adjoint operator $K$ mapping $L^2(\mathbb{R}^d) \rightarrow L^2(\mathbb{R}^d)$ as $f \rightarrow \int K(x,y) f(y) d\mu(y)$, let us define its eigenvalues $\lambda_i$ and eigen-...
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