# Questions tagged [integral-operators]

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### Schur test for integral operators - almost everywhere strictly positive function?

I need to apply the Schur test (https://en.wikipedia.org/wiki/Schur_test), but would like to do so with a test function which vanishes on the boundary of the space (in fact, it must necessarily, as ...
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### An Orthogonality Problem of Eigenfunctions of homogeneous Fredholm equation

Suppose we have a integral equation $$\int_{-1}^1 \frac{\text{sin }c(x-y)}{\pi (x-y)}\psi(y)dy=\lambda \psi(x),\quad|x|\le1.$$ By the Fredholm equation theory, we know that this equation has ...
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### Gradient of square norm in RKHS

Consider a RKHS $\mathcal H$, with continuous reproducing kernel $K$. I am confused regarding the gradient of $\tfrac 12 \Vert \cdot \Vert_{\mathcal{H}} ^ 2$. On the one hand, I'd expect it to be ...
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### Eigenvalues and eigenfunctions of an integral operator

Let $T$ be an integral operator with kernel $K(x,y)=e^{|x-y|}$ on $L^2(-1,1)$. How can we find the eigenfunctions and eigenvalues of $T$? Even though I am not sure whether the following arguments are ...
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### Point spectrum of an integral operator

Let we have $$Tu(x) = \cfrac{1}{x}\int_0^x u(y)dy$$ so that $u \in L^2(0,1)$. How can I show that $(0,2) \subset \sigma_p(T)$ and $T$ is not compact?
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### Are Covariance Operators based on square integrable stochastic Processes semi-positive definite?

Given a $\textbf{square integrable stochastic process}$ $X$ with $E\left(X\left(t\right)\right)=0$ $\forall t$ the $\textbf{Covariance Operator}$ is defined by \begin{align} C_X: L^2 \rightarrow L^...
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### A “convolution”-like operator for “moving difference” of functions?

What is the following operator called? $$(f\star g)(t) = \int_{-\infty}^\infty \|f(\tau)-g(\tau-t)\|dt$$ I am thinking it can peehaps be built as convolution of exponential and/or logarithmic ...
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### Integral Operator bounded on $L^p$, or infinite-dimensional matrix operator bounded on $\ell^p$

Note: As a newer user, forgive me if this is against forum etiquette to bring attention to old (one of which is unanswered) posts. To make this question more novel to the site, I expound on why I did ...
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### Calculation/Verification of an integral kernel for $\operatorname{e}^{t\Delta}(1-\Delta)^{-\frac{1}{4}}$

Given the operator $T = \operatorname{e}^{t\Delta}(1-\Delta)^{-\frac{1}{4}} \colon L^p(\mathbb{R}^3) \to L^p(\mathbb{R}^3), p \in (1,\infty), t > 0$ I want to calculate its kernel $K_t$ in order to ...
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### Weak-* convergence in $C^1([-1,1])^*$

Given the sequence of functionals $$f_n(x)=\int_{\frac{1}{n}\leq|t|\leq1}\frac{x(t)}{t}dt$$ in $C([-1,1])$ respectively $C^1([-1,1])$. How can I show that $(f_n)$ weak-*-converges in $C^1([-1,1])^*$...
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### $\alpha \in L(L^p, L^{2p})\; \Rightarrow\; \alpha \in L(L^{2p}, L^{4p}), \, p \in [1,\infty)?$

Let $p \in [1,\infty)$ and $\alpha \in \mathcal{L}(L^p\!, \,L^{2p}),$ meaning $\alpha\colon L^p \rightarrow L^{2p}$ is linear and bounded, where $L^p$ and $L^{2p}$ stand for the $L^p$- and $L^{2p}$-...
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### Fractional Laplacian

Q(1) It is well known that if $s \rightarrow 1$ then the fractional Laplacian converges to the classical Laplacian but the form of the Laplacian still remains in the non-local form whereas it is known ...
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### Resolvent set of Volterra integral operator

Let $V : L^2(0,1) \to L^2(0,1)$ be given as follows $$Vu(x)=\int_0^x{u(t) dt}$$ We know that $\sigma(V)=\{0\}$. How to find $(V-\lambda I)^{-1}$ when $\lambda \neq 0$? I tried to find it at least ...
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### Find linear operator given set of eigenvalues

The condensed problem: I have a bounded, compact, self-adjoint, linear operator $A$ on $L^2([a,b];\mathbb{R})$ with positive eigenvalues $\{\frac1{\lambda_i}\}_i$. Let $\lambda > 0$. Is there an ...
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### Compact integral operator on $C([0,1])$

On space $X=C([0,1])$ we are given an operator $T:X \to X$ with $$Tf(x)=\int_0^x{f(y)}dy$$ I proved using Arzela-Ascoli that the operator is compact. The other question is to prove that $T(B_X)$ is ...
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