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12
votes
4answers
5k views

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
2answers
346 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
2answers
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
1answer
363 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
1answer
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
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 ...
5
votes
1answer
214 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 <\frac{...
4
votes
1answer
197 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 \...
4
votes
0answers
116 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 R}...
3
votes
0answers
61 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. ...
2
votes
1answer
123 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 - \sum\...
2
votes
1answer
255 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
1answer
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& \end{...
2
votes
1answer
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
1answer
188 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
votes
0answers
31 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 $0<\alpha<1$. ...
2
votes
0answers
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
1answer
144 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
votes
0answers
134 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
0answers
215 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} r(x,\xi)...
1
vote
1answer
138 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 f(x^{-1}\...
1
vote
2answers
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
vote
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)=\...
1
vote
0answers
60 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
vote
0answers
82 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
0answers
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
vote
0answers
227 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 (\zeta-...
0
votes
2answers
827 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
1answer
124 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
votes
0answers
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
votes
0answers
37 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
votes
0answers
43 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
votes
0answers
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
votes
0answers
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
votes
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 ...
0
votes
1answer
264 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 non-...
0
votes
0answers
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
votes
0answers
137 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
votes
0answers
381 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: $$||v||_{L^{2}(\Sigma)}...