Questions tagged [integral-kernel]
The integral-kernel tag has no usage guidance.
0
votes
0answers
32 views
Integrability green function
This is maybe a trivial question but i need some clarification to make it clearer in my mind.
Consider the fundamental solution of the equation $ \partial_{t} u - \partial^{2}_{xx}u=0$ given by the ...
2
votes
0answers
36 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
1answer
182 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
$$
\begin{align}
dX_{U(N)} & = \frac{1}{N!(2\pi)^N}
\begin{vmatrix}
1 & 1 ...
0
votes
0answers
57 views
Solving an equation with a definite integral
Are there any techniques, either analytical or numerical, to solve an equation of the following form:
$$f(x)=\int_{-\infty}^\infty g(x,y)h(y)\,dy$$
where $f(x)$ and $g(x,y)$ are known functions, and ...
4
votes
1answer
125 views
Which utility functions are linearly transformed by normal perturbations?
I'll ask this question as pure economics, as pure math, and showing the translation.
Economics (micro):
Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...
6
votes
1answer
185 views
Injectivity of a Fredholm operator
While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory.
Question : Suppose $K$ is ...
4
votes
2answers
211 views
Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold
I'm trying to close in on a definitive answer to my own question BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?, and think I ...
1
vote
0answers
90 views
Existence theorems Volterra Equation of second kind on unbounded domains
The general Volterra Equation of the second kind in convolution form can be described by:
$$
\phi(x) = \int_a^x K(x-t)\phi(t)\, \mathrm{d}t + f(x), \text{ for } x\geq a
$$
Suppose we wish to ...
1
vote
0answers
45 views
Outer product $\sum_i |k_{x_{i}}(\cdot)\rangle\langle k_{x_{i}}(\cdot)|$ of reproducing kernel functions as identity operator in RKHS?
In a separable Hilbert space $\mathcal{H}$, given a complete orthonormal basis $\{|e_i\rangle\}$, the identity operator can be written as $\mathbb{1} = \sum_i |e_i\rangle\langle e_i|$. Now if this ...
4
votes
0answers
190 views
Is there any orthogonal basis set that satisfies an integral equation appearing in physics?
I am considering the following integral equation
$$
f(x) = \int_{-W}^W \frac{\exp(-x y)}{\cosh(y)} g(y) dy,
$$
where $-1 \le x \le 1$, $f(x)$ and $g(y)$ are real functions.
This integral equation ...
0
votes
0answers
92 views
kernel integration reference needed
I have asked this question in MS, but get no answer.
I am working on a problem which lead me to the following integral, given $h:R \rightarrow R$ is a nonzero smooth function with compact support. For ...
3
votes
1answer
235 views
When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)
Let $L$ be an elliptic linear operator on $\mathbb R^n, n\geq3$. For simplicity, let's stick to the following Schrodinger operator
$$
Lu:=-\Delta u+V(x)u
$$
where $V\geq0$ is the electric potential, ...
1
vote
0answers
55 views
basis representation of a special sinc kernel
I have a functional map from $(x,y) \in \mathbb{R}^2$ to another function $f_{x,y}(z,w) \in \mathbb{C}$. (Variables $z,w $ range from $-\infty$ to $\infty$.) That is, for any pair $x,y$, I get a ...
2
votes
1answer
341 views
Gaussian bounds on Dirichlet heat kernel
Let $(M, g)$ be a compact Riemannian manifold and let $p(t, x , y)$ be the heat kernel of $M$. Then there exist constants $c, C > 0
$ such that $$\frac{c}{t^{n/2}}
e^{-\frac{1}{4t}d(x, y)^2} \leq ...
1
vote
1answer
83 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 $0<\alpha<1$.
...
2
votes
0answers
218 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{...
3
votes
0answers
116 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 ...
1
vote
1answer
124 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 ...
2
votes
1answer
133 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 ...
5
votes
2answers
262 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-...
5
votes
1answer
336 views
Every self-adjoint trace class operator on $L^2$ has integral kernel
I have asked this question on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...
0
votes
0answers
88 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 $\...
0
votes
0answers
89 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 ...
1
vote
0answers
114 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}.$$
Now, if $R(y_1,...
4
votes
1answer
327 views
Hardy-Littlewood-Sobolev inequality using fractional sobolev norm on the RHS
Using Hardy-Littlewood-Sobolev inequality, we can prove that:
$$\left| \int_0^1\int_0^1 |x-y|^{-\frac{1}{2}} f(x)f(y) \mathrm{d}x \mathrm{d}y \right| \leq C \left\| f \right\|_{L^{4/3}(0,1)}^2 \leq C \...
1
vote
1answer
107 views
Fredholm integral with functions constrained to [0;1]
I am trying to feed information about the solution when solving an inverse problem given by a Fredholm integral of the form
$$
g(t)=\int_{a}^{b}K(t,s)f(s)ds.
$$
Say I know $g(t)$ and $K(t,s)$, and ...
0
votes
1answer
310 views
Integral Transform with associated Legendre Function of second kind as kernel
In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, $Q^{\...
1
vote
0answers
125 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 ...
3
votes
1answer
196 views
Nuclearity properties of Integral operators with Schwartz type kernels
Consider an integral operator $$T:L^2({\mathbb R}^n)\to L^2({\mathbb R}^n),\qquad (Tf)(x)=\int_{\mathbb R^n} dy\,K(x,y)f(y),$$ with Schwartz type kernel $K\in{\mathscr S}({\mathbb R}^{2n})$ (smooth ...
3
votes
0answers
629 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{...
1
vote
2answers
521 views
Improper integral $\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$
In my research, I ran into following types of improper integral
$\int^\infty_0 e^{-a x^2} \cosh (b\sqrt{1+x^2})$
with real parameters $a>0,b>0$.
Mathematica cannot evaluate them. It also ...
4
votes
0answers
234 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 ...
3
votes
3answers
158 views
tranforms that lowers the number of variables of a function
Is there any linear map that lowers the number of variables of functions, namely a map that maps a function of several variables to functions of one variable and at the same time the original ...
0
votes
2answers
160 views
Analyticity of Logarithmic Integrals
Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
4
votes
0answers
218 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}
...
8
votes
0answers
199 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 ...
0
votes
2answers
465 views
Poisson inequality for subharmonic functions
This is probably a very basic matter, but I am looking for a proof of the Poisson inequality for subharmonic functions, which reads
$$\varphi(r \mathrm{e}^{\mathrm{i} \theta})\leq\frac{1}{2\pi} \...
2
votes
1answer
318 views
The relation between the heat kernel on the principal bundle and the heat kernel on the base manifold
This question is mainly about Section 5.2 of the book "Heat Kernels and Dirac Operators" by Berline, Getzler and Vergne.
Let $M$ be a compact Riemannian manifold without boundary and $P\rightarrow M$ ...
1
vote
0answers
422 views
A problem about Joint sine and cosine fourier transform
There is a problem on Page 60 of Book of B.M.Budak, A. A. Samarskii and A. N. Tikhonov, whose title is A Collection of Problems in Mathematical Physics (New York, Dover, 1964). The problem (the ...
3
votes
2answers
667 views
Schwartz kernel theorem for topological spaces
Is there some regularizing version of Schwartz kernel theorem for topological spaces, i.e., in the form of
Every continuous linear map $A\colon C\prime(X_2) \to C(X_1)$ is given by a kernel $k \in ...
7
votes
2answers
496 views
Asymptotic Expansion of the Schrödinger kernel?
My stackexchange post [http://math.stackexchange.com/questions/275830/schrodinger-kernels-on-manifolds] was somewhat unsatisfactory (also because I may not have stated clear enough what my interest ...
7
votes
1answer
420 views
Convergence of Schwartz kernels implies convergence of operators
Let $K$ be a smoothing operator on $\mathbb{R}^n$, i.e., it defines a map on all Sobolev spaces $K\colon H^r(\mathbb{R}^n) \to H^s(\mathbb{R}^n)$ for all $r, s \in \mathbb{R}$. Now (a variation of) ...
5
votes
2answers
480 views
Closed formula for heat kernel
Is there, similar to the Mehler kernel, a closed formula for the heat kernel of the heat equation associated to the Laplacian
$$ -\sum_j \frac{d^2}{dx_j^2} + 2\sqrt{-1} \sum_j \lambda_j \frac{d}{dx_j} ...
3
votes
2answers
1k views
Techniques to solve equations involving a definite integral [closed]
Are there any well known techniques to solve a problem of the following form: $$\int_a^b f(x,\alpha) dx = g(\alpha),$$ where $a,b\in\mathbb{R}$ are fixed, $f$ and $g$ are known functions, $\alpha\in\...
1
vote
1answer
350 views
Fredholm Integrals of the Second Kind with an unknown Kernel
I am trying to solve the equation:
$\phi(x)=\int_{-\infty}^\infty K(x, t)\phi(t)dt$
for $K$ given $\phi$. This closely resembles a Fredholm Integral of the Second Kind, which has the form:
$\phi(x)...
3
votes
2answers
680 views
Do these kernel functions satisfy the semi-group property?
Dear Friends,
Define the kernel functions for $a\ge 1$,
$$
G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in R\;,
$$
where the constant $C_a$ is some normalization ...
6
votes
4answers
7k views
What does “kernel” mean in integral kernel?
In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc.
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
