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7 votes
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Injectivity of a Fredholm operator

Surprisingly (to me), the statement is false. My counterexample is a little messy, but the idea is fairly simple. Take $T = 1$ and set $a_n = \frac{1}{n}$ and $b_n = 1 - \frac{1}{n}$ for $n \in \...
Nik Weaver's user avatar
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5 votes
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Is the solution to the Fredholm integral equation of the first kind a continuous analogue of Cramer's rule for matrix equations?

Denote the Fredholm determinant $\Delta=\det(I+K)$, then the solution of the Fredholm equation $g=(I+K)f$ is given formally by $$f=(I+K)^{-1}g=\frac{1}{\Delta}\left(\frac{\partial\Delta}{\partial K}\...
Carlo Beenakker's user avatar
5 votes

Function orthogonal to $|y-x|$ on $[0,1]$ for every $y \in [0,1]$?

No, linear combinations of functions of the form $|x-y|$ are dense in $C([0, 1])$ (because you can approximate any continuous function by the piecewise-linear), so (if $f\in L^1(0,1)$, otherwise the ...
Aleksei Kulikov's user avatar
5 votes

Eigenfunctions and eigenvalues of an operator defined by a certain integral

The inverse operator is $L^{-1} = 1/2 \cdot (x-1)^{-1} d^2/dx^2 - 1/2 \cdot (x-1)^{-2} d/dx$. Its eigenfunctions are derivatives of Airy functions, $Ai^{\prime } ((2\lambda )^{1/3} (x-1))$, $Bi^{\...
Michael Engelhardt's user avatar
4 votes
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integral kernel function for the SU(N) group

The integral kernel for ${\rm U}\,(N)$, due to Dyson, has been generalized by Katz and Sarnak to other compact groups (Random Matrices, Frobenius Eigenvalues, and Monodromy, page 121). Their result ...
Carlo Beenakker's user avatar
4 votes

Injectivity of an integral operator

Your operator $K$ is a Hilbert-Schmidt operator since its kernel belongs to $L^2$. As a result this is a compact operator whose spectrum contains a sequence of eigenvalues $\\{\lambda_k\not=0\\}$ with ...
Bazin's user avatar
  • 15.5k
4 votes

General strategy for studying the decay of eigenvalues of kernel integral operators

Birman and Solomyak have studied this question quite intensivly. The paper may not be the easiest to understand, but it does cover in a very general setup, what regularity conditions on the kernel ...
Paul Pfeiffer's user avatar
4 votes
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How to find the inverse of a product of two integral equations

Let $F(y) = \int_0^y \rho(x) dx$. Then your equation reduces to: $$R(y) = F(y)(F(l) - F(y))$$ Note that for any value of $F(y)$, we get that $R(y) \leq \frac{F(l)^2}{4}$; this is easily proven by ...
user44191's user avatar
  • 4,991
4 votes
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Rate of convergence of Fejer kernel to the Dirac delta function

For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$. Two explicit ...
Carlo Beenakker's user avatar
4 votes

Calculation of an inverse Mellin transform

Because of the identity $$\frac{d}{dz}\, _1F_1(b;b-a;z)=\frac{b }{b-a}\, _1F_1(b+1;b+1-a;z)$$ the function $K(x)$ is given by $$K(x)=-\frac{(b-a)\Gamma(a)}{b\Gamma(b)}f(a,b,x)$$ with $f(a,b,x)$ the ...
Carlo Beenakker's user avatar
4 votes
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Solution set of integral equation/ Kernel of linear operator

I think it's easiest to work in the Hilbert space setting for this problem, i.e., to consider $F$ is a functional on the space $L^2([0,1]^2)$, where $[0,1]^2$ is endowed with the Lebesgue measure. Let ...
Jochen Glueck's user avatar
4 votes
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Is the heat kernel of a manifold $p$-integrable?

Let $g(\cdot):=h(t, x, \cdot)$. For all $p\ge 1$, you have the elementary inequality $$ \left( \int_M |g|^p\right)^\frac1p=\left( \int_M|g||g|^{p-1}\right)^\frac1p \le \left(\int_M |g|\right)^\frac1p\...
Giuseppe Negro's user avatar
3 votes
Accepted

Existence of integral kernel

If $f(x,y)=\sum_i c_i \chi_{E_i \times F_i}(x,y)=\sum_i c_i \chi_{E_i}(x) \chi_{F_i}(y)$ with $(E_i\times F_i)\cap (E_j \times F_j) =\emptyset$ for $i\neq j$, define $$\phi (f)=\sum_i c_i \int_{R^d} (...
Giorgio Metafune's user avatar
3 votes
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Numerical methods for integral eigenvalue equation

To solve $\int d^3 y\, K(x,y,\lambda) f(y) = f(x)$ I would discretize the coordinates $x\mapsto x_n$, $y\mapsto y_m$, $K(x,y,\lambda)\mapsto K(x_n,y_m,\lambda)\equiv K_{nm}(\lambda)$ and solve the ...
Carlo Beenakker's user avatar
3 votes
Accepted

On an integral equation

The answer is no. A counterexample is $$ f(t,x) = x - \frac{1}{2} -\frac{1}{24} t^2 $$ $$ B(t,s,x) = \left( x-\frac{1}{2} \right) (t-s) $$ (Method: I obtained this by expanding $f$ and $B$ into power ...
Michael Engelhardt's user avatar
3 votes
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Inverting convolutions over finite intervals

A modification of the Wiener-Hopf method for this type of problems is described in Convolution equations on finite intervals and factorization of matrix functions and in Finite interval convolution ...
Carlo Beenakker's user avatar
3 votes

Gaussian bounds on Dirichlet heat kernel

The answer to the question is yes for the upper bound (albeit with different constants in the gaussian factor), and no for the lower bound. For the upper bound, and for any domain $\Omega \subset M$, ...
Raziel's user avatar
  • 3,183
2 votes
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Error estimate in the spectral theorem of compact operators on a Hilbert space

I'm following Szego's book on orthogonal polynomials. In chapter III he considers $f \in L^2\left((a,b),d\alpha \right)$, with $-\infty \leq a < b \leq \infty $. There exist a set of orthogonal ...
Amir Sagiv's user avatar
  • 3,554
2 votes
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Is there a way to solve this integral equation?

There is no unique solution to the integral equation $$\int\limits_0^\infty g(\kappa, x_0) \exp{\left[-\frac{\left(\xi + \kappa \right)^2}{2\alpha\theta}\right]} d\kappa % = \exp{\left[-\frac{\left(\...
Carlo Beenakker's user avatar
2 votes
Accepted

Trace-class properties of integral operator

$k$ being compactly generated, you can as well assume that $k$ is a smooth function defined on $\mathbb{T}^2$ and $Op(k)$ acts on $L^2(\mathbb T)$ (for $\mathbb{T} = \mathbb R/\mathbb Z$ the unit ...
Mikael de la Salle's user avatar
2 votes

Which utility functions are linearly transformed by normal perturbations?

$\newcommand{\al}{\alpha} \newcommand{\de}{\delta} \newcommand{\De}{\Delta} \newcommand{\ep}{\varepsilon} \newcommand{\ga}{\gamma} \newcommand{\Ga}{\Gamma} \newcommand{\la}{\lambda} \newcommand{\Si}{\...
Iosif Pinelis's user avatar
2 votes
Accepted

Gradient condition implies Hörmander condition

I figured out proving it by using mean value theorem (thanks to @WillieWong). Observe that \begin{align*} \int_{|x|>2|y|}|K(x-y)-K(x)|dx &\leq \int_{|x|>2|y|}|\nabla K(tx+(1-t)(x-y))||y| dx\\...
Lev Bahn's user avatar
  • 239
2 votes
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When does an inverse PDE operator have a kernel (i.e. a fundamental solution?)

Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel ...
Bazin's user avatar
  • 15.5k
1 vote

Structure of the inverse of a Fredholm integral operator of the second kind

Not really an answer, but some remarks: Even if the spectral radius of $K$ is less than $1$, there are counterexamples that the resolvent need not have the required form: This is related to the fact ...
Martin Väth's user avatar
  • 1,869
1 vote

Unique solution for 2$\times$2 Fredholm integral equations system

The operator $f\to K(x) \int_0^1 f(s) ds$ is a compact operator from $C([0,1];\mathbb R ^n)$ equipped with the sub norm into itself, if $K$ is an $n\times n$ matrix valued function for any finite $n\...
username's user avatar
  • 2,464
1 vote

Boundedness of integral operators on spaces of continuous functions

Too long for a comment. Your requirement is too stringent and it is quite likely that to get continuity from $L^\infty$ into itself, it is indeed necessary to have $$ \text{esssup}_x\int\vert k(x,y)\...
Bazin's user avatar
  • 15.5k
1 vote
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Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold

Thanks to Jochen Wengenroth's comments, I can now give the full answer: the idea is that $C_\mathrm{c}^\infty(M) \hookrightarrow L^2(M, \mathrm{d} \mu_g) \xrightarrow{G} L^2(M, \mathrm{d} \mu_g) \...
JahvedM's user avatar
  • 119
1 vote

Gaussian bounds on Dirichlet heat kernel

I think you will find what you need here: http://ac.els-cdn.com/002212369090106U/1-s2.0-002212369090106U-main.pdf?_tid=c2567084-37e6-11e6-a2ed-00000aab0f6b&acdnat=...
Michael Renardy's user avatar
1 vote

Well-definedness for a singular integral

Assuming the function $f$ of class $C^1$, you find that, using Taylor's formula with integral remainder, $$ f(t)-f(s)=(t-s) f_1(t,s),\quad \text{with $f_1$ continuous},$$ so that $ (T_\alpha f)(t)=\...
Bazin's user avatar
  • 15.5k
1 vote
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Various limits of the Christoffel Darboux Kernel

It seems that both questions receive some answers in this paper. In th $L^2$ norm, Section 9 seems to stipulate convergence and a limit $K$. For the uniform and pointwise convergence, we have on p. ...
Amir Sagiv's user avatar
  • 3,554

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