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9 votes
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No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?

Yes it is possible. Let $K\in S'(\mathbb{R}^2)$ be the distribution acting on test functions $f(x,y)$ by $$ K(f)=\int_{|x-y|\ge 1} \frac{f(x,y)}{|x-y|}\ dx\ dy \ + \int_{|x-y|< 1} \frac{f(x,y)-f(x,...
Abdelmalek Abdesselam's user avatar
8 votes
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Cauchy path integral as a linear operator: kernel and image?

Such an integral over a simple (non-closed) curve is called the Cauchy type integral. It is convenient to define $$F(z)=\frac{1}{2\pi i}\int_\gamma \frac{f(\zeta)d\zeta}{\zeta-z},\quad z\not\in\gamma.\...
Alexandre Eremenko's user avatar
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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7 votes

A numerical calculation for an integral

Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; ...
Carlo Beenakker's user avatar
5 votes

Delta-distribution composed with a function from the Fourier representation

What you want to do is the pull-back of distributions. And there is a theorem (cf. Hörmander 1, Theorem 8.2.4) that if the set $\{(\phi(x),\eta) \colon \phi'(x) \eta = 0\}$ and $\operatorname{WF}(\...
mcd's user avatar
  • 591
5 votes

No kernel of the form $\lvert x - y\rvert^{-1}$ on tempered distributions?

(Edit: see below... I misread the question...) One particular way to regularize that integral, relating to meaningful things, is by viewing it as Hilbert transform applied to one Schwartz function, ...
paul garrett's user avatar
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5 votes
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Infinite dimensional version of a simple fact on certain singular matrices

For the first question, the answer is not necessarily. Very rough idea: The rank-nullity theorem doesn't always hold on infinite dimensional spaces. Rough idea: Let the operator $A$ be defined on $...
Willie Wong's user avatar
  • 38.7k
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
5 votes

Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

Maple does not know a symbolic answer for this. The very special case $a=2,b=0,c=1,d=2$ is evaluated by Maple in terms of the Whittaker M function or a $\;{}_1F_1$ hypergeometric function: $$ \int_{0}...
Gerald Edgar's user avatar
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5 votes
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Is there a solution to $\int_{\theta-1}^{x} \left(\frac{y+1-\theta}{\theta} \right)^{-\frac{1}{\epsilon+3}} y^{-\frac{\epsilon+4}{\epsilon+3}}dy$?

With some effort (the lower integration limit requires care) I found this answer for the definite integral: $$I(x)=\int\limits_{\delta}^{x} \left(y-\delta\right)^{-\frac{1}{ \epsilon+3}} y^{-\frac{\...
Carlo Beenakker's user avatar
5 votes
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Why is it difficult to define a direct integral of Banach spaces or Banach algebras?

To see the problems one faces, start with the simplest example of a family of Banach spaces, the constant family $x\mapsto B$ with $x\in X$ and $X$ some measurable space. The direct integral will be ...
G. Rodrigues's user avatar
  • 1,848
4 votes

Infinite dimensional version of a simple fact on certain singular matrices

Willie Wong answered the general case, and I'd like to give a counterexample for the symplectic case: Let $M=S^2$, the standard 2-sphere embedded in $\mathbb{R}^3$ as the unit sphere, with the ...
kosta's user avatar
  • 375
4 votes

Can an integral equation always be rewritten as a differential equation?

Well, the answer, if there is any, must be subtle. Take the following example. Let $H$ be the Hilbert transform. Then $$\frac{d}{dx}Hu=f$$ is an integral equation, where the kernel is the Fourier ...
Denis Serre's user avatar
  • 51.9k
4 votes
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Fourier transform with cubic exponential

Do you mean the Airy transform? A more recent reference is The Airy transform and the associated polynomials (2010). In your notation the function $F_3$ is the Airy transform of the Fourier ...
Carlo Beenakker's user avatar
4 votes
Accepted

Calderon-Zygmund theorem for the kernel of spherical harmonics

Your operator $T$ is a Fourier multiplier with symbol $m = \mathcal{F}[Y^m_k/r^n]$; that is, $\mathcal{F}[Tu]=m \mathcal{F}[u]$. It is a relatively simple exercise to show that the norm of $T$ on $L^...
Mateusz Kwaśnicki's user avatar
4 votes

Eigenvalues of an integral operator

If $\lambda=1$ is an eigenvalue, then $\int_0^1 K(s,x)u(s) ds=0$ for the corresponding eigenfunction. Hence a condition to rule this out is that $\{K(s,x),x \in (0,1)\}$ spans a dense subspace of $L^2$...
Michael Renardy'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
  • 15.7k
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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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

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

Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

A closed form expression in terms of a special function exists for $b=0$, when $$\int_0^y x^{-a} \exp \left(-\tfrac{1}{2} c^2 x^{-2d} \right) \,dx=\frac{1}{2 d}y^{1-a} E_{1-\frac{a-1}{2 d}}\left(\...
Carlo Beenakker's user avatar
4 votes

Integral operator (compactness)

Integrals over strongly continuous functions with values in the compact operators are compact. That's a result by Jürgen Voigt.
Jochen Glueck's user avatar
4 votes
Accepted

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
3 votes

A Fredholm equation with a particular kernel

$\newcommand{\la}{\lambda} $ Let us formalize the question as follows: Take any complex number $\la$. Let $K(s,t):=s\wedge t-st$ for $s,t$ in $[0,1]$. Find $x\in L^2[0,1]$ such that \begin{...
Iosif Pinelis's user avatar
3 votes

A Fredholm equation with a particular kernel

You can immediately translate this integral equation into an easy second order linear ODE on $[0,1]$ with boundary conditions $x(0)=0$, $x(1)=1$. (I'm not going to do it for you). Just note that your ...
Pietro Majer's user avatar
  • 58.1k
3 votes
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Coupled partial differential and integro-differential equation

I am going to elaborate on Nemo's comment. You first need to note that $\frac{d^2}{dx^2}e^{|x|} = 2\delta_0(x)\cdot e^{|x|} + e^{|x|}$ in the sense of distributions, where $\delta_0(x)$ represents the ...
Sanal's user avatar
  • 46
3 votes
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Regularity of solutions to certain integral equation

Let's first make the interval equal to $(-1,1)$ by the change of variable $$ x = \frac{a+b}{2} + \frac{b-a}{2}\, t ; $$ then the equation becomes $$ \int_{-1}^1 u(s)[c+\log |s-t|]\, ds = g(t) , \quad\...
Christian Remling's user avatar
3 votes

BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

The book by Melrose was almost exactly written to address issues like the ones you mentioned above, where the very first example is the one in the question. The construction fo the parametrix is the ...
Bombyx mori's user avatar
  • 6,219
3 votes
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BVPs for elliptic PDOs: When do Green functions ($L^2$ inverses) define pseudo-differential operators in the interior?

By complementing Deane Yang's strategy with Jochen Wengenroth's observations in the related question Composition of a smoothing operator with an $L^2$-bounded operator, non-compact Riemannian manifold,...
JahvedM's user avatar
  • 119
3 votes
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Solution to $\int_{0}^{y} x^{-a} \exp \left[- \frac{(b - cx^{-d})^2}{2} \right] dx$

By using series expansion, change of variable and Eq. (3.381.9) of the book: "I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 8th ed. Burlington, MA, USA: Academic ...
Felipe Augusto de Figueiredo's user avatar

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