# Questions tagged [integral-operators]

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### 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 ...
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### Lower-bound for $\inf_{f \in H} \|\nabla f\|_{L^2(\tau_d)}/\|f\|_H$ for an RKHS $H$ induced by exponential kernel on sphere

Let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $\tau_d$ be the uniform probability measure thereupon. Let $H_K \subseteq L^2(\tau_d)$ be an RKHS of square $\tau_d$-integrable functions, ...
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### Seek help to formalize an argument to positiveness of function defined inductively by integral [closed]

I have on domain $[0,\infty)$ a known and positive function $f(x)$ and two unknown functions $g(x), h(x)$ that start positive when $x=0$. I also know that if $h(x)$ is positive, then $g(x)$ is also ...
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### 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 ...
1answer
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### Solution set of integral equation/ Kernel of linear operator

I am interested in understanding the solutions $\phi$ of the following integral equation: $$0=\int_0^1 \int_0^1 \phi(x,y)(x-y)\thinspace dx\thinspace dy.$$ Equivalently, I am interested in ...
1answer
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### Conditions for continuity of an integral functional

Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F_f$, defined ...
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### Injectivity of an integral operator

Consider the operator $$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$ with $k\in L^2((0,1)\times(0,1)).$ I want to know under what assumption the kernel is reduced to zero. i....
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### 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 ...
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### Eigenvalues of an integral operator

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
1answer
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### A numerical calculation for an integral

I am interested in the numerical calculation of $$F(\eta)=\frac{2}{π}\int_0^{+\infty}\sin(t\eta-t^3)\frac{dt}{t}\quad\text{for \eta\ge 0}.$$ I believe that the function $F$ is bounded, but I do ...
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### 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 $\lambda\in\mathbb R$, ...
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### Numerical methods for IDE [closed]

I would like to read a popular literature on the topic "Numerical methods for integro-differential equations". Could you recommend me any articles or book with a brief overview of some methods (maybe ...
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### 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 $\mathbb{R}^d$....
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### Which utility functions are linearly transformed by normal perturbations?

I'll ask this question as pure economics, as pure math, and showing the translation. Economics (micro): Which utilities have the property that whenever $EU(X)>EU(Y)$, the same is true after ...
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### 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 ...
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### Injectivity of a Fredholm operator

While doing my study on the boundary-crossing time of a stochastic process, I happened to deal with the following question which is somehow related to Fredholm theory. Question : Suppose $K$ is ...
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### 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 ...
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