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Questions tagged [integral-kernel]

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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 ...
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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
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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, ...
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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 ...
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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 ...
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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 $0<\alpha<1$. ...
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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{...
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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 ...
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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
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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 ...
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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
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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 ...
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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 $\...
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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 ...
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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
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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
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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
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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^{\...
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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
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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 ...
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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{...
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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 ...
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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
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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 ...
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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 ...
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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} ...
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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
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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} \...
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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 $P\rightarrow M$ ...
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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 ...
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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 ...
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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
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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
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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} ...
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2answers
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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
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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
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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 ...
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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 ...