Questions tagged [integral-kernel]
The integral-kernel tag has no usage guidance.
45
questions with no upvoted or accepted answers
10
votes
0
answers
581
views
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}
\...
- 101
8
votes
0
answers
220
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
5
votes
0
answers
359
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, ...
- 51
4
votes
0
answers
236
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 ...
4
votes
0
answers
301
views
Convergence of Schwartz Kernels
I read this question, and I would like to ask the opposite: Assume that I have a sequence of smoothing operators $(P_n)$ with (hence smooth) kernels $(p_n)$ converging strongly to some smoothing ...
- 9,591
4
votes
0
answers
285
views
Solving a Fredholm equation with a piecewise kernel : Karhunen-Loeve of a stopped Brownian motion
Is there a way to solve analytically the Fredholm integral equation of the second kind
$$
\int_0^{100} K(s, t) f(s) ds = \lambda f(t)
$$
where the kernel has the piecewise 'linear' form
\begin{align}
...
user24451
3
votes
0
answers
162
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
$$\omega=\...
- 1,055
3
votes
0
answers
243
views
Extending Ky Fan's eigenvalues inequality to kernel operators
--Migrating from MSE since it might fit better here--
Base result
The following result in Terry Tao's book, 'p. 47, Ky Fan inequality' reads as:
$$\sum_i\lambda_i(A+B) \leq \sum_i \lambda_i(A) + \...
- 245
3
votes
0
answers
168
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 $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
- 6,726
3
votes
0
answers
108
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 ...
- 5,038
3
votes
0
answers
191
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 ...
- 3,554
3
votes
0
answers
1k
views
What is the inverse kernel of this integral transform?
I am looking for the associated inverse kernel to the integral transform $T$ defined by
$(Tf)(u) = \int_{-\infty}^{+\infty} K(u,t)f(t) \ dt,\ \ u \in \mathbb{R^+}$
whose kernel is $K(u,t) = \frac{...
- 31
2
votes
1
answer
403
views
Solving $\psi_{xxx} + (u(x) - (ik)^3))\psi = 0$ for $x > 0$, $k \in \mathbb C$, and $u(x)$ smooth
I was reading Initial-Boundary Value Problems for Linear PDEs with Variable Coefficients by P. Treharne and A. S. Fokas, when I came across the following ODE formulated as part of a Lax pair for a ...
- 547
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},\...
- 91
2
votes
0
answers
265
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}...
- 156
2
votes
0
answers
112
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
0
answers
57
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 ...
- 545
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 ...
- 545
2
votes
0
answers
248
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:
$$\lambda \begin{pmatrix} X_1(u) \\ X_2(u) \end{...
1
vote
0
answers
59
views
A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
vote
0
answers
123
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 ...
- 996
1
vote
0
answers
54
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 $\psi_h(x)$...
1
vote
0
answers
77
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 ...
- 1,230
1
vote
0
answers
79
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
$$S(t)S(s)u(x) = S(t+s)u(x)....
- 585
1
vote
0
answers
53
views
Properties of a kernel convolution $K'(x,y) = \int_X\int_X K_0(x,a)K(a,b)K_0(b,y)d\mu(a)d\mu(b)$ where $K$ and $K_0$ are kernels on $(X,\mu)$
Let $(X,\mu)$ be a probability measure space and $K:X \times X \to \mathbb R$ be a (psd) kernel on $X$. Let $K_0$ be another kernel on $X$ and defined a new kernel $\widetilde K$ on $X$ by
$$
\...
- 6,726
1
vote
0
answers
45
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, ...
- 11
1
vote
0
answers
95
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 ...
- 201
1
vote
0
answers
67
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 $\...
- 6,726
1
vote
0
answers
63
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 $\geq1$. ...
- 1,266
1
vote
0
answers
102
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
89
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
$$Tu = \int_{\mathbb{R}^{d}} e^{i\varphi(x,y)/h} a(x,y) u(x) dx, \ \ a \in \mathcal{C}_{c}(\mathbb{R}^{2d}), \ \ \varphi \in \...
- 31
1
vote
0
answers
148
views
Existence of continuous integral kernel
Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$).
A ...
- 31
1
vote
0
answers
67
views
Angle between Fleming-Viot type 3-particle system
Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
- 284
1
vote
0
answers
109
views
Existence theorems Volterra Equation of second kind on unbounded domains
The general Volterra Equation of the second kind in convolution form can be described by:
$$
\phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a
$$
Suppose we wish to ...
- 31
1
vote
0
answers
100
views
Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?
In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
- 11
1
vote
0
answers
71
views
basis representation of a special sinc kernel
I have a functional map from $(x,y) \in \mathbb{R}^2$ to another function $f_{x,y}(z,w) \in \mathbb{C}$. (Variables $z,w $ range from $-\infty$ to $\infty$.) That is, for any pair $x,y$, I get a ...
- 31
1
vote
0
answers
125
views
Transformation of kernel
I have the following problem at hand.
Define the kernel
$$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$
Now, if $R(y_1,...
- 21
1
vote
0
answers
172
views
Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity
I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
- 11
1
vote
0
answers
437
views
A problem about Joint sine and cosine fourier transform
There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
- 11
0
votes
0
answers
73
views
the design of kernel function and integral transform
I read a solution of an integral inequality.
The solution uses condition $$f(1)=f(0)=f'(0)=0$$ to derive that
$$f(x)=\int_0^1k(x,y)f'''(y)dy$$, $$k(x,y)=\begin{cases}-\frac{x^2(1-y)}{2} & x\leq y\...
- 39
0
votes
0
answers
47
views
Relation between Kernel density estimation and Reproducing kernel Hilbert space?
The procedure of kernel density estimation using a kernel $K$ is very similar to the construction of an RKHS from the kernel $K(x,y) = K(x-y).$ However, this viewpoint is not mentioned every place I ...
- 1,641
0
votes
0
answers
55
views
Is polynomial interpolation with RKHS in some way more advantageous than simple Lagrange interpolation?
[Question originally posted here but maybe it is more suitable for this site.]
The reproducing kernel Hilbert space associated with the polynomial kernel $K(x,z)=(1+xz)^{d-1}$ (or other similar ...
- 1,641
0
votes
0
answers
69
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 $g$...
- 631
0
votes
0
answers
127
views
Kernel for projection operators onto the spaces of piecewise linear loops
For every $m\in\mathbb{N}_+$, $k=0,\dots,m-1$, denote $I_{m,k}:=(\frac{k}{m},\frac{k+1}{m})$.
Denote $W = H^1(S^1,\mathbb R^{2n})$ = The Hilbert space of $C^1$ maps maps $v(t)$ from the circle to $\...
- 1,873
0
votes
0
answers
107
views
Convergence of the solution of Volterra integral equation with convergent kernel (reposted, need help!)
Consider the following Volterra integral equation
$g(t)=∫_0^tK_n(t,s)w_n(s)ds$
where $g(t)$ and $K_n(t,s)$ are continuous and $K_n(t,s)≥K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ converges to ...
- 243