# Questions tagged [integral-kernel]

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86
questions

10
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0
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### Eigenfunctions of the integral kernel $1/(x^2 + x'^2)$

My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...

3
votes

1
answer

234
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 ...

2
votes

1
answer

142
views

### Rate of convergence of Fejer kernel to the Dirac delta function

This seems like something one might find in a book so I would be grateful for any references you think may be helpful.
I am interested in the rate at which of a function integrated against the $N$th ...

2
votes

1
answer

64
views

### Numerical methods for integral eigenvalue equation

I have an integral equation which is not exactly an eigenvalue type equation, but similar:
$$\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$$
Here $\lambda$ can be thought of as an eigenvalue, so it is ...

1
vote

1
answer

155
views

### On an integral equation

Let $B: C^{\infty}([0,1]^3)$ satisfy
$$B(t,t,x)=0 \quad \text{for all $t,x \in [0,1]$.}$$
Let $f \in C^{\infty}([0,1]^2)$ satisfy the following integral equation:
$$ \int_0^1 f(t,x)\,dx + \int_0^t\...

1
vote

0
answers

118
views

### Heat kernel coefficients for Laplacian in instanton background

The heat kernel coefficients $b_{2k}(x,y)$ of the covariant Laplacian in an $SU(2)$ instanton background (for simplicity let's say $q=1$ topological charge, so the 't Hooft solution) on $R^4$ is ...

1
vote

0
answers

41
views

### $L^2$ norm of a kernel with a variable width

Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...

2
votes

0
answers

43
views

### When can convolutional integral operators be sampled

Consider an integral operator $F:C^{1/2}([0,T],\mathbb{R}^n)\rightarrow \mathbb{R}$ of the form
$$
f\mapsto \int_0^T f(t)^{\top}\kappa(T-t)dt,
$$
for some $\kappa\in C^{\infty}(\mathbb{R})(\mathbb{R},\...

0
votes

0
answers

57
views

### Construct RKHS $H$ on $\mathbb R^d$ s.t for every $C \gt 0$, there exists $f_0 \in H$ with $\|f_0\| = 1$ and $\|\nabla f_0\|_{L^2(\gamma_d)} \gt C$

Fix a positive integer $d$, and let $\gamma_d$ be the standard Gaussian measure on $\mathbb R^d$.
Question. Construct (or prove non-existence of) a reproducing kernel Hilbert space (RKHS) $H \subseteq ...

1
vote

0
answers

74
views

### On an integral equation of Volterra type

Consider the following integral equation
$$f(t) = A(t) + \int_0^t K(s,t)f'(s)ds,\quad \forall t\ge 0,\label{1}\tag{$\ast$}$$
where $A: \mathbb R_+\to [0,1]$ and $K: \{(s,t): 0\le s\le t\}\to[0,1]$ are ...

2
votes

1
answer

322
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];\...

1
vote

0
answers

65
views

### 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)....

1
vote

0
answers

50
views

### 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
$$
\...

1
vote

0
answers

41
views

### 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, ...

3
votes

0
answers

145
views

### 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=\...

1
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0
answers

70
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 ...

2
votes

0
answers

191
views

### 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) + \...

5
votes

0
answers

290
views

### 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, ...

3
votes

0
answers

143
views

### 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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...

3
votes

1
answer

276
views

### 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 ...

4
votes

1
answer

355
views

### Calculation of an inverse Mellin transform

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

1
vote

1
answer

139
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
\...

0
votes

1
answer

216
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 ...

1
vote

1
answer

176
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 $...

1
vote

0
answers

62
views

### 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 $\...

2
votes

1
answer

523
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 ...

4
votes

1
answer

220
views

### 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 ...

1
vote

0
answers

62
views

### 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 $\geq1$. ...

3
votes

1
answer

273
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 ...

1
vote

0
answers

98
views

### 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 ...

1
vote

0
answers

75
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

1
answer

556
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....

5
votes

1
answer

551
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 ...

3
votes

0
answers

102
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 ...

1
vote

0
answers

129
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

1
answer

507
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 $$ Lf(x):=\int_0^1 k(x,y)f(y)dy$$. ...

1
vote

0
answers

63
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-}^...

2
votes

0
answers

251
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### 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}...

2
votes

0
answers

108
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### 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

437
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)...

0
votes

0
answers

67
views

### 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 $g$...

2
votes

0
answers

53
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 ...

2
votes

0
answers

44
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 ...

4
votes

1
answer

273
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 ...

3
votes

1
answer

275
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### 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 ...

6
votes

1
answer

810
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 ...

6
votes

1
answer

756
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 ...

1
vote

0
answers

109
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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 ...

1
vote

0
answers

78
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### 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 ...

4
votes

0
answers

220
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 ...