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21 views

Integral kernel connecting dual differential equation solutions?

Consider a partial differential operator $D_{\{p\}}(x)$ in several variables $x=(x_1,x_2,...,x_n)$ in $\mathbb{R}^n$ and a set of a few parameters $\{p\}$. The solution $F_{\{p\}}(x)$ such that $$D_{\...
0
votes
1answer
131 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 ...
2
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0answers
34 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 $0<\alpha<1$. ...
1
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0answers
57 views

Jacobi's Elliptic functions - Kernel

I have a kernel expressed in terms of Jacobi's elliptic functions. In particular I have four functions $$\begin{eqnarray} &\rho_{11}(u,v)&=\frac{1- k sn u sn v}{sn u-k sn v},\nonumber\\ &\...
1
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1answer
99 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 ...
1
vote
1answer
88 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 ...
4
votes
2answers
222 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-...
2
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0answers
163 views

Boundedness of the solution of the integral equation associated to the heat kernel

Let $\Omega$ be a bounded open set of $\mathbb{R}^n$ ($n\geq 2$), $\Omega\in C^{1+\alpha}$ with $\alpha\in(0,1)$. Let $\Phi_n$ be the fundamental solution of the heat equation, that is $$\Phi_n(t,x)\...
4
votes
1answer
110 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
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0answers
44 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
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0answers
171 views

Minimization of nonlinear integral operator

See also on MSE. For non-negative self-adjoint traceclass operators $0\leq T \leq 1$ with $\mathrm{tr}T^\alpha=N$ on the Hilbert space $L^2(\mathbb{R}^3)$ s.t. $\operatorname{tr}(-\Delta T^\alpha)=\...
0
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0answers
85 views

Singular integral equation

Investigating a control problem for heat equation I stacked on solution of this integral equation which seems to be singular: $$ \int_0^1\mathcal{K}(\tau)u(\tau)d\tau=\mathcal{M}, $$ in which $\...
0
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0answers
93 views

Existence of a solution to $xf(x) = \int_0^1 k(x,y) yf(y) dy$

Crosspost - I asked very similar question on math.sx. Let $X = (0,1)\times (0,1)$ and $k\colon X \to \mathbb{R}$ be a Lebesgue measurable non-negative function such that $$ \int_0^1 k(x,y) dy = 1$$ ...
0
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0answers
46 views

The property reservation conditions in the functional iteration process

Given a integral equation: $$K(y,p)=\int_{a}^{b}f(x,y)K(x,p)dx$$ Using the iteration method,choosing an arbitrary inition function $K^{0}(x,p)$: $$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$ ...
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0answers
53 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
105 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
200 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
73 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
173 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^{\...
0
votes
0answers
87 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
147 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
231 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
475 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
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0answers
176 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 ...
2
votes
3answers
155 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
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2answers
153 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
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0answers
175 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} ...
6
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0answers
174 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
421 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
287 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
385 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
568 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 ...
6
votes
2answers
388 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
285 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) ...
4
votes
2answers
425 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
263 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
559 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
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4answers
5k 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 ...