Questions tagged [integral-operators]

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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 ...
Alex's user avatar
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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}...
Gandalf Lechner's user avatar
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230 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 $\...
Bazin's user avatar
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4 votes
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Roots of smoothing operators

Suppose that $(M,g)$ is a smooth, compact Riemann manifold and $K:M\times M\to\mathbb{R}$ is a smooth, symmetric nonnegative function. We regard is as the Schwartz kernel of a smoothing operator. In ...
Liviu Nicolaescu's user avatar
4 votes
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218 views

Lower semi-continuity of integration

I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))\,dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$....
ABIM's user avatar
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Asymptotic form of the eigenvalues / eigenfunctions of a Fredholm equation

Consider the kernel: $$k(x,y,\xi) = \sqrt{\frac{1}{\pi} \frac{1}{y+a}} \exp \left(-\frac{\xi^2}{y+a }\right)$$ I am trying to find the asymptotic form of the solutions to the following homogeneous ...
valle's user avatar
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Reference request: trace norm estimate

In a paper I am currently reading, the author uses that if $T$ is an operator given by the kernel $$T(x,y) = \int_{\mathbb R} p(x,z) q(z,y) dz,$$ then $$\lvert \operatorname{tr} T \rvert \leq \lVert T ...
Staki42's user avatar
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139 views

Extrapolated Integral operator (compactness)

I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a ...
Malik Amine's user avatar
3 votes
0 answers
197 views

Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$. Does there exist a nonzero operator $X $ satisfying the following system $VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator? $$ V(f) (x) =\...
Anas Abbas H.'s user avatar
3 votes
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113 views

Schatten norm estimate of spatially truncated resolvent of Laplacian

Consider the operator $-\Delta$ on $L^2(\mathbb R^d)$. In my studies I stumbled upon operators of the form $$1_{\Gamma_n} (-\Delta -z)^{-1} 1_{\Gamma_m},$$ where $1_{\Gamma_m}$ denotes multiplication ...
user271621's user avatar
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96 views

Boundedness of Calderon-Zygmund type operator

I am trying to prove the following fact. Suppose $\varphi\in C_c^\infty(\mathbb{R}^1)$. Define $$T(u)(x):=P.V.\int_{\mathbb{R}^1}\frac{\varphi(x)-\varphi(y)}{|x-y|^{\frac32}}u(y)dy$$ where P.V. means ...
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Formal way to prove existence and continuity in an integral equation

In the paper "A Boundary Value Problem Associated with the Second Painlevé Transcendent and the Korteweg-de-Vries Equation" by Hastings and McLeod, the authors study the ODE $$\frac{\mathrm{d}^2 y}{\...
user1337's user avatar
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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. ...
JT_NL's user avatar
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Existence of solutions to n-dimensional integral equation with solutions into [0,1]

I have a research problem I am working on where a step involves proving the existence of solutions to a certain kind of integral equation. A statement of this problem is as below. I would appreciate ...
anshad's user avatar
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Derivative of a functional involving integral and level set

Let $\Omega$ be a bounded smooth domain. For $u\colon \Omega \to \mathbb{R}$, define the functional $$F(u) = \int_{\{u=0\}}g(x) \; \mathrm{d}x$$ where eg. $u \in H^2(\Omega) \cap C^0(\bar\Omega)$ and ...
Blah000's user avatar
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0 answers
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Translation request: Boundedness of Cauchy integral on Lipschitz boundary

The reference: "L'intégrale de Cauchy définit un opérateur borné sur $L^2$ pour les courbes lipschitziennes" (https://annals.math.princeton.edu/1982/116-2/p04) is written in French. Can we ...
Yidong Luo's user avatar
2 votes
0 answers
130 views

Optimization of functionals with constraints

I have a minimization problem as follows: $\min\left( \int_0^1\int_0^1\beta(t)\beta(s)G_1(t, s)dtds\right)^{1/2}+\left( \int_0^1\int_0^1\beta(t)\beta(s)G_2(t, s)dtds\right)^{1/2} $ $\texttt{s.t.}\;\;\;...
Shuoyang Wang's user avatar
2 votes
0 answers
125 views

Asking for results on critical points and similar properties of solutions of nonlinear Volterra integral equations - Physically coherent solutions

I have a system of nonlinear Volterra integral equations of form $$x(t)=x_0+\int_0^t K(t,s)F(x(s))ds$$ and I am interested on the critical points of $x(t)$, I mean maximum, minimum, increasing and ...
Quiet_waters's user avatar
2 votes
0 answers
67 views

Unique continuation for integral operator

I accidentally met such question. Let's start from easy ones. Let $\Omega$ be an open convex domain in $\mathbb{R}^2$ and $u(x)$ satisfies that $$u(x) = \int_{\partial\Omega} \nabla_y G(|x-y|)\cdot ...
Yimin's user avatar
  • 143
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258 views

Modified variational formulation of heat equation

The heat kernel $u:\mathbb{R}^n\times (0,\infty)$ is defined as the solution to $$ u_t = \Delta u, $$ subject to certain boundary conditions and can alternatively be described, in variational form, as ...
ABIM's user avatar
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2 votes
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575 views

When the square root of integral operator becomes also integral operator (with continuous kernel)?

Let $X$ be a compact metric space and $\mu$ be strictly positive Borel measure on $X$. Let $T:L^2(X,\mu)\rightarrow L^2(X,\mu)$ be a self-adjoint, compact, and positive operator on the Hilbert space $...
S.Lim's user avatar
  • 449
2 votes
0 answers
150 views

Run-away Volterra operator

For a continuous function $k:[0,1]^2\to \mathbb{R}$, let $A$ be the generalized Volterra integral operator on $C([0,1],\mathbb{R})$ defined by $$ A(f)(t)\triangleq \int_0^t k(s,t)f(s) ds, \quad t \in [...
ABIM's user avatar
  • 5,019
2 votes
0 answers
128 views

Limit involving singular kernel: $\lim_{s\to 1}(1-s)\int_{\Omega}\frac{(u(x)-u(y))}{|x-y|^{d+2s}} d y. $

Let $\Omega\subset \Bbb R^d$ be a bounded $C^1$ domain. Let $u:\Bbb R^d\to \Bbb R$ be a function in $C^2_b(\Bbb R^d)$. I would like to compute the following limit: for $x\in \partial \Omega$ $$L= \...
Guy Fsone's user avatar
  • 1,033
2 votes
0 answers
112 views

Solving Fredholm integral equation in Lp

I have a very simple integral equation $$ f(x) - \lambda \int_a^be^{x-y}f(y)dy=1 $$ which is Fredholm of the 2nd kind, with separable kernel. It is needed to find the values of $\lambda\in\mathbb R$, ...
jonathan wolf's user avatar
2 votes
0 answers
57 views

Integral equation with kernel defined in a rectangle

Let us consider $$f(x) + \lambda \int_0^4 {K(s,x)f(s)ds=0} ,{\text{ x}} \in {\text{(0}}{\text{,1)}}$$ Observe that the kernel is not defined on a square. My question: Can I apply the classical ...
Gustave's user avatar
  • 545
2 votes
0 answers
44 views

A special integral equation of Volterra type

Let $a,f \in L^2(0,t)$ (where $t \leqslant 1$), and consider the following integral equation: $$ f(t)\int_0^t a(s)\,ds + \int_0^t a(t - s) f(s) \, ds = 0 $$ My question is : under what condition ...
Gustave's user avatar
  • 545
2 votes
0 answers
87 views

Functional equation involving integrals and exponential

Can we find on $\mathbb{R}^+$ a real positive function $f(x)$ (in $C^{\infty}$) such that: $$\int_0^{\infty} f(x) e^{\lambda \int_1^{x} f(t)^2 dt} dx=0$$ where $\lambda$ is a complex number (with $0&...
Bertrand's user avatar
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2 votes
0 answers
79 views

One-dimensional integral equation uniquely solvable?

I recently met a question similar to this one and I would like to post it here, because I basically found nothing: We define the (possibly unbounded) integral operator $T:D(T) \subset C_0(\mathbb{R}) ...
BaoLing's user avatar
  • 319
2 votes
0 answers
503 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 ...
Ali's user avatar
  • 37
2 votes
0 answers
77 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 $$\...
BigM's user avatar
  • 1,543
1 vote
0 answers
48 views

About Fourier integral operators

Consider the operator $$(A_\Delta\psi)(p) = \int_\Delta \int_{\mathbb{R}^n} e^{ix\cdot(p-k) + i\phi(x,p,k)} a(x,p,k) \psi(k) dk\: dx$$ where $\Delta$ is a Borel or Lebegue set in $\mathbb{R}^n$, in ...
V. Moretti's user avatar
1 vote
0 answers
23 views

Characterization of the Picard's condition for integral equation

Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...
Mingzhou Liu's user avatar
1 vote
0 answers
109 views

When is the solution to a Fredholm integral equation a PDF?

I have two questions about inhomogenous Fredholm integral equations of the first kind: $$f(x) = \int_a^b K(x,t) g(t) dt$$ where $f, K$ are known and $g$ is not. If a unique solution for $g$ exists, ...
Wilbur's user avatar
  • 111
1 vote
0 answers
58 views

Reference request: normal trace and the conormal derivative associated to the operator $Div (A \nabla)$ for a symmetric positive definite $A$

Let $A$ be a $3\times 3$ symmetric positive definite matrix. I am looking for a reference where I could find in which sense the normal trace $\gamma$ and conormal derivative $\gamma_n$ associated to ...
SAKLY's user avatar
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1 vote
0 answers
70 views

Convolution Integral Equation on a compact subset of the real line

I am dealing with the following equation: $$ f(x) = g(x) + \intop_{X} dt K(x-t)f(t) \;,\qquad \left\lbrace \begin{array}{c}f(x)>0\;,\;x\in X \\ f(x)<0\;,\;x\notin X \end{array}\right.$$ where $X$...
Stefano's user avatar
  • 105
1 vote
0 answers
154 views

Bifurcation points in parametric Hammerstein Integral equation

I am a theoretical physicist working on integrable systems, hence I preemptively ask forgiveness for the eventual lack of mathematical rigor. My question concerns the properties of a particular ...
Stefano's user avatar
  • 105
1 vote
0 answers
88 views

How does the constant bound on Hormander's $L^2$ estimate for non-degenerate phase depend on the cut-off function?

A classical estimate, due to Hormander, assets that the integral operator $$Tu = \int_{\mathbb{R}^{d}} e^{i\varphi(x,y)/h} a(x,y) u(x) dx, \ \ a \in \mathcal{C}_{c}(\mathbb{R}^{2d}), \ \ \varphi \in \...
Yilin Ma's user avatar
1 vote
0 answers
164 views

Polar decomposition of the Volterra integral operator

Repost of this Math.SE question due to a lack of answers (No one was able to help me find the closed form of $U_T$ and $|T|$ after two bounties). I also searched extensively online but couldn't find ...
ViktorStein's user avatar
1 vote
0 answers
148 views

Existence of continuous integral kernel

Let $(X_i,\mathcal{F}_i,\mu_i)$, $i\in \{1,2\}$, be a $\sigma$-finite measure space and $E_i$ an ideal of measurable functions on $X_i$ with full carrier (for example $E_i=L^p(X_i,\mu_i)$). A ...
MathWorker's user avatar
1 vote
0 answers
67 views

Angle between Fleming-Viot type 3-particle system

Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
maliesen's user avatar
  • 284
1 vote
0 answers
114 views

Uniqueness of solution of Volterra Integral Equation with deviating argument

In the context of a physics problem, I am looking at a linear integral equation 2nd kind Volterra equation with deviating (centrosymmetric) argument in the unknown $u(t) \in L^2[a,b]$: \begin{equation}...
Nebojša Đurić's user avatar
1 vote
0 answers
128 views

Reference for Existence and uniqueness of an Integro-Differential Equation

I have an Integro-Differential Equation (IDE) of the following form: $$ x'(t) = f(t,x(t)) + \int_0^t K(t-s, x(s), x(t)) ds, $$ I have found this classical reference, but the IDEs considered therein ...
Darkwizie's user avatar
  • 121
1 vote
0 answers
129 views

Characterisation of functions for which the Fourier transform commutes with a particular operator

Defining the operator $\phi$ by: $\phi(f(x))=\frac{1}{|x|} f(\frac{1}{x})$, and noting $\mathcal{F}$ the Fourier transform on the real line, can we characterize all the functions (with real variable ...
Bertrand's user avatar
  • 1,111
1 vote
0 answers
198 views

Interpretation of Smoothing Operators as $\Psi$DO's

In most of the computations I've seen involving pseudodifferential operators (referred to as $\Psi$DO's from now on) we do not have true equality. Instead we often only have equivalence of operators, ...
Patch's user avatar
  • 367
1 vote
0 answers
70 views

Reference request: Explicit solution of Fredholm integral equations of the second kind

I am looking for explicit solutions of the following Fredholm integral equation of the second kind $$ \phi(t) = 1 + \int_0^1 G(|t-s|) \phi(s) ds, \qquad t \in [0,1], $$ for specific kernels $G$, e.g. ...
Elias Strehle's user avatar
1 vote
0 answers
110 views

Smoothing property of integral operators

Consider an integral operator $J$ with kernel $k(x,y)$ (assuming its properties are nice), can we describe the operator $J$ in $$Jf(x) := \int_0^1 k(x,y) f(y) dy$$ as an isomorphism(i.e. $J$ has a ...
newbie's user avatar
  • 319
1 vote
0 answers
125 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, $K_n(t,s)$ ...
user155214's user avatar
1 vote
0 answers
371 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-...
Hans's user avatar
  • 2,169
0 votes
0 answers
12 views

Understanding relation of 2 dependent, integral equations which are nested in a Bayesian Expectation

I'm trying hard to try understand the recursive nature between two equations in a recent macroeconomics paper, but my question mainly relates to how mathematically such recursive equations can depend ...
Justin Lee's user avatar
0 votes
0 answers
143 views

Why is this function in $L^1$?

I had a question about a claim made in the paper "Group Invariant Scattering" and why it is true. Consider the function $h_j(x) = 2^{nj}\psi(2^jx)$, where $\psi$ is a function such that $\...
Bobo's user avatar
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