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Questions tagged [integral-operators]

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3
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1answer
130 views

An explicit example of a compact integral operator in one dimension to help with intuition?

I am about to start studying compact operators and I always find the best way to get intuition on a new area is to start with a simple example. So what would be a an example of a compact integral ...
1
vote
1answer
209 views

Eigenvalues and Eigenfunctions of a Particular Self Adjoint Operator.

Consider the operator $T(f)(x): L^2[0,1] \rightarrow L^2[0,1]$ defined $$T(f)(x)=\int_{0}^{1-x} f(y) \ (1-y-x) \ dy.$$ (Assume $L^2[0,1]$ is the set of square integrable real valued functions over the ...
0
votes
1answer
68 views

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 ...
7
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0answers
616 views

Integral operator is bounded on $L^p$ if it maps $L^p$ to itself

Here is a homework excercise. Let $(X,\Omega,\mu)$ be a $\sigma$-finite measure space,$1\leq p <\infty.$ and suppose that $k:X\times X\rightarrow \mathbb{C}$ is an $\Omega \times \Omega$ ...
4
votes
0answers
54 views

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 ...
4
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0answers
108 views

Maximal ideals of closed algebra generated by Volterra operator

Let $V$ denote the Volterra integral operator on $L^2[0, 1]$ defined by $$ Vf(s)=\int_0^s f(t) dt $$ and let $A$ be the closed subalgebra of $\mathcal{B}(L^2[0,1])$ generated by $V.$ Show that $A$ ...
4
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0answers
111 views

Is this integral operator on $L_2(0, \infty)$ compact?

Let's define $T:L_2(0,\infty) \to L_2(0,\infty)$ as $$(Tf)(x) = \int_0^\infty \frac{f(y)\sqrt{xy}}{x^2y^2+1}dy.$$ I'm interested, if this operator is compact. $T$ is integral operator with kernel $K = ...
3
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0answers
142 views

Singular value decomposition of a specific integraloperator

I want to determin the singular value decomposition of the integral operator $$L^2(0,1) \to L^2(0,1) ; f \mapsto Af(\cdot) = \int_0^\cdot f(y)dy.$$ Its adjungate is given by $$A^*f(\cdot) = \int_\...
3
votes
0answers
54 views

Is the operator $Af = \int_0^1 k(x,y) f(x) dx$ irreducible provided that $Af = f$ has unique solution such that $\int_0^1 f = 1$

Let $k \in L^1((0,1) \times (0,1))$ be non-negative and such that $$ \int_0^1 k(x,y) dy = 1, \quad x \in (0,1). $$ Let $A\colon L^1(0,1) \to L^1(0,1)$ be defined by $$ Af(y) = \int_0^1 k(x,y) f(x) dx, ...
3
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0answers
47 views

Computing the asymptotic spectrum of a negative distance kernel

Consider the following integral operator: $$K(f) : x \mapsto\int_0^1 K(x,x')f(x') dx', \quad \text{where} \quad K(x,x') = - |x-x'|^{3/2}.$$ The kernel is sometimes referred to as a negative ...
2
votes
0answers
59 views

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 ...
2
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0answers
207 views

Norm of an integral operator

I have an exercise that I need to solve and I can't finish it. Let $k \in \mathcal{C}([0,1] \to \mathbb{R})$. Proove that this operator : $$ \begin{array}{ccccc} T & : & \left(\mathcal{C}([0,1]...
2
votes
0answers
134 views

Kernel of Integral operator

Let $H: L^2(M) \longrightarrow L^2(M)$ be a bounded operator. Here, $M$ can be a Riemanniannian manifold, or some open subset of $\mathbb{R}^n$. Question: What can I say about the Schwartz Kernel $k$ ...
2
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0answers
98 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is real-...
1
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0answers
20 views

Soft Question: Highlights of Osculatory Integral Theory

I'm curious to look into Oscillatory integral operator theory and was wondering what are some of the highlights, main results, and historical development. Are there distributional characterizations ...
1
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0answers
90 views

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 ...
1
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0answers
191 views

Finding the nullspace and range of an integral operator

I'm trying to determine the nullspace and range of the following integral operator, but I'm having trouble proceeding. Let $K:C([0,1])\to C([0,1])$ be defined by $$Kf(y)=\int_{0}^1 \sin(\pi(x-y))f(y)...
1
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0answers
151 views

Diagonal of kernel

Suppose that $K:L^2([0,1],\mathbb C)\to L^2([0,1],\mathbb C)$ is an integral operator given by $$ Kf(x)=\int_0^1k(x,y)f(y)dy $$ for each $f\in L^2([0,1],\mathbb C)$ and $x\in[0,1]$. In general, the ...
1
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0answers
57 views

Infinite application of an operator to all coordinates

Consider a pair of functions $f:\mathbb{R}^n \rightarrow \mathbb{R}$ and $p:\mathbb{R}^2 \rightarrow \mathbb{R}$ and the following integral: $g(x_1,x_2,\dots,x_n) = \int_{\mathbb{R}^n} f(t_1,t_2,\...
1
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0answers
141 views

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 ...
1
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0answers
328 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
1
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0answers
119 views

Maximization of a convolution

Given a piecewise smooth, bounded, integrable, etc., causal function $h(t)$, such that $h(t)=0$ if $t<0$, my question is which bounded, causal, piecewise smooth, etc., function $c(t)$ will maximize ...
1
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0answers
39 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in C^{\...
1
vote
0answers
57 views

Showing that a certain operator maps to $\mathscr{C}([0,1])$

I'm considering the operator $T$ given by (Tf)(x)=$\int_0^1k(x,y)f(y)dy$ with $dom(T)=\mathrm{L}^1([0,1])$, where $k\in\mathscr{C}([0,1]^2)$ and want to proof that it maps to $\mathscr{C}([0,1])$. I ...
1
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0answers
71 views

How to solve this special case of Fredholm Integral Equation of the First Kind

General form of 'Fredholm Integral Equation of the First Kind' $f(x) = \int_a^b{K(x,t)\phi(t)} dt$ Where $\phi(t)$ is the unkown My special case is $1 = \int_a^b{k(t)\phi(t)} dt$ A trivial ...
1
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0answers
119 views

Integro-differential eigenvalue problem

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace h(x',y')\nabla'f_n(x',y')\big\rbrace}{\sqrt{(x-x')^2+(y-y')^...
1
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0answers
85 views

Proof Check for Compactness of Integral Operator

Above is my question. I have completed the question, but I'm not 100% about my proof for the final part - it seems like I haven't done enough. I've shown that if $U$ and $V$ are compact, then so is $...
1
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0answers
47 views

Does an inequality between kernels imply an inequality between the norms of integral operators?

Assume that $g(x,y)$ and $h(x,y)$ are two positive functions such that $0<g<h$ and assume that $$T_g, T_h : L^2(B^n,R)\to L^2(B^n,R)$$ are integral operators defined by $$T_k[f](x)=\int_{B^n} f(...
0
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0answers
9 views

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 ...
0
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0answers
18 views

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 ...
0
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0answers
45 views

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 ...
0
votes
0answers
38 views

How to express the equation $D|f\rangle=0$ as a differential equation?

Consider the Eqn. $$D|f\rangle=0.\tag{1}$$ I want to express it as a differential equation $$\mathcal{D}_x f(x)=0\tag{2}$$ where $\mathcal{D}_x$ is the differential operator representation of the ...
0
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0answers
95 views

Eigenvalue of Integral Operator and Gamma Function

$''$ Prove that the following integral operator $ Ku(x) = \int_{0}^{ \infty } \ e ^{-xy} u(y) dy $ has as eigenfunction the $ φ_α(x) = \sqrt {Γ(α)} x^{-α} + \sqrt {Γ(1-α)} x^{α-1} $ for $ ...