The integral-operators tag has no usage guidance.

**12**

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**4**answers

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

Given an integral equation is there always a differential equation which has the same (say smooth) solutions?
It seems like not but can one prove this in some example?
Edit: Naively I'm hoping for ...

**8**

votes

**2**answers

343 views

### Convergence of an oscillatory integral

Given $f\in H^1(\mathbb{R}^3)$ and $t>0$, consider the following integral:
$$I_f(t):=\int_{\mathbb{R}^3}\int_0^{+\infty}e^{-s+i\frac{(s+|x|)^2}{t}}f(x)dsdx$$
I need to show that $I_f(t)$ is finite, ...

**8**

votes

**2**answers

592 views

### Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate ...

**7**

votes

**1**answer

353 views

### Fredholm operators in $K$-theory?

Do Fredholm operators show up in K-theory? Why or why not? The idea of infinite Grassmannians classifying vector bundles is pretty straightforward, but why would adding in additive inverses and what ...

**6**

votes

**1**answer

1k views

### Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite.
We are interested in ...

**6**

votes

**0**answers

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 ...

**5**

votes

**1**answer

212 views

### Estimate of incomplete binomial integral

Let $0\le k \le n$. Prove that
$$
n\binom{n}{k}\int_{0}^{\frac{k}{n+1}}t^k(1-t)^{n-k}\,dt \le 1/2.
$$
As far as I know
1) it is proved for $\frac{k}{n+1}\le 1/2$ and
2) not proved for $1/2 ...

**4**

votes

**1**answer

192 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 ...

**3**

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**0**answers

60 views

### Eigenvalues of approximations to product-convolution operators

Consider an operator $T: L^2 \mapsto L^2$ of the form $TA = g (h \ast A)$ where $g$ is and $h$ are bounded $C^\infty$ functions.
This operator $T$ can be shown to be Hilbert-Schmidt, hence compact. ...

**3**

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**0**answers

112 views

### Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...

**2**

votes

**1**answer

120 views

### Minimizing a convex integral function

Consider the following constrained optimization with the integral objective function
$$
\min_{x_i\in D} \int_{t_1}^{t_2} \frac 1 {t - \sum\limits_{i = 1}^N x_i f_i (t)} \, dt
$$
where $t - ...

**2**

votes

**1**answer

250 views

### Interpolation between weighted $L^p$ spaces

Let $K:\mathbb{R}^3\backslash\{0\}\times\mathbb{R}^3\backslash\{0\}\rightarrow\mathbb{C}$, such that $K(x,y)=K(y,x)$ and $K(x,y)=|x|^{-1}|y|^{-1}H(x,y)$, with $H$ locally bounded.
Let $T$ be the ...

**2**

votes

**1**answer

145 views

### Solvability of a Fredholm system in $L^2$

Suppose $\lambda\not=0\in\mathbb{C}$. Does the following system have a non trivial solution in $L^2 [0,1]$?
\begin{array} {lcl} \int_0 ^1 f(y)\log|x-y|dy=\lambda f(x) \\\int_0 ^1f(x)dx=0& ...

**2**

votes

**1**answer

77 views

### Post composition of integral

Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...

**2**

votes

**1**answer

183 views

### Kernel of the Radon transform

Consider the following generalized version of the Radon transform. Let $X,Y,Z$ be compact smooth manifolds. Let $p\colon Z\to X$, $q\colon Z\to Y$ be smooth maps. Let $m$ be a fixed smooth density ...

**2**

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**0**answers

67 views

### Regarding a result of I.Vekua on integral equations of first kind

Suppose $k\in L^2([0,1]\times[0,1])$ is a symmetric kernel. Presumably the following is true(under the assumption that $k$ is weakly singular):
Either
$$\displaystyle\int_0^1f(y)k(x,y)dy=0$$or
...

**2**

votes

**1**answer

141 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 ...

**2**

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**0**answers

132 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**2**

votes

**0**answers

212 views

### Schwartz kernel of a pseudodifferential operator with singular symbols

If we have a smooth symbol $r(x,\xi)$ of order $d$ the corresponding pseudodifferential operator $P$ is integral operator with kernel given by
$$K(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi} ...

**1**

vote

**1**answer

131 views

### Orbital integral for matrix coefficients

I am currently aiming at estimating orbital integrals. Maybe surprizingly, I hope for some help in the compact case (ramified places), in proving the usual formula
$$O_\gamma(f) = \int_G ...

**1**

vote

**2**answers

182 views

### Double Integral of plane wave squared over a circular domain

Hello everyone! I need to calculate the following integral:
$\int\int_{\sqrt{x^2+y^2}\leq a}\cos^2(v_x x+v_y y)dxdy$
First step, I convert to polar coordinates:
$\int_0^a\int_0^{2\pi}\cos^2(v_x ...

**1**

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**0**answers

168 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 ...

**1**

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**0**answers

58 views

### Volterra integral equation of the first kind

I would like to know whether I can find a unique solution for F to the following problem $$\int_0^t F(g(u)(t-u)) du = h(t)$$ where both g and h are known and "nice" (in fact, I can make any ...

**1**

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**0**answers

69 views

### Solving Fredholm Integral Equations of the first kind

I am looking to solve a Fredholm Integral Equation of 1st kind of the form:
$\int_{a}^b h(u-u')g(u')du' = f(u)$
where $h$ is a general function.
I know from the Hilbert-Schmidt theorem that all I ...

**1**

vote

**0**answers

90 views

### Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation
$$
g(t) = \int_0^t K_n(t,s)w_n(s) ds
$$
where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...

**1**

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**0**answers

224 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty ...

**0**

votes

**2**answers

810 views

### Inequality of Lebesgue integral with $L^p$-norm

Let $X_t(\omega)$ be a continuous function $t\rightarrow L^p(\omega)$ (i.e., if we fixed the variable $t$ we obtain a function which belongs to $L^p$), with $t\in[0,T]$ and $\omega\in\mathbb{R}$.
I ...

**0**

votes

**1**answer

123 views

### Solvability of an integral equation

Is it possible to find $f,g\in L^2[(0,1)\times(0,1)]$ such that $$\log|x-y|=\int_0^1f(x,t)g(t,y)dt\:\:\:\forall x,y\in(0,1).$$

**0**

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**0**answers

71 views

### “increasing” the logarithmic energy of certain measures

Let $0<a<b<1$ and $f\in L^2[0,a]$ be a real-valued function with $\int_0^af^2=1.$ Define its logarithmic energy by $$\mathcal{E}_a(f)=\int_0^a\int_0^af(x)f(y)\log\frac{1}{|x-y|}dxdy$$
Q. ...

**0**

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33 views

### On the continuity of Riemann-Liouville integral

For a function $f:(0,1)\to\mathbb{R}$, the Riemann-Liouville integral of $f$ is defined by
$$
(I^{1-\nu}f)(t):=\frac{1}{\Gamma(1-\nu)}\int_{0}^{t}\frac{f(s)}{(t-s)^{\nu}}ds,
$$
where $\nu\in(0,1)$ is ...

**0**

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41 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**

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**0**answers

57 views

### $L^2$-boundedness of integral operator

Let $a:{\bf R}^d\to M^{d\times d}$ semi-definite matrix consisted of smooth functions i.e.
$$
\langle a(x) \xi,\xi \rangle=\sum\limits_{k,j=1}^d a_{kj}(x)\xi_j \xi_k \geq 0, \ \ x\in {\bf R}^d, \ \ ...

**0**

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45 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 $K^{0}(x,p)$:
$$K^{1}(y,p)=\int_{a}^{b}f(x,y)K^{0}(x,p)dx$$
...

**0**

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**0**answers

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 ...

**0**

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**1**answer

255 views

### solution uniqueness of non-linear Fredholm equations

the equation is
$F(x)=G\left(\int k(x,y)f(y)dy\right)$ $(*)$
where $f(x)=\frac{dF(x)}{dx}$ is unknown and it's required to be non-negative. With integration by parts we'll have the form of a ...

**0**

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47 views

### A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators
(in the context of Arthur-Selberg trace formula)?

**0**

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133 views

### can we generalize the cauchy principal value and hadamard's finite part for multiple integrals $ n>1 $

I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one
My question is when can we (or if negative answer why can not ) generalize the ...

**0**

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**0**answers

369 views

### Notation for a functional L2 matrix norm

Hi,
Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation:
...