# Questions tagged [integral-operators]

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33 questions
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 ...
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 ...
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 ...
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 $\Omega \times \Omega$ ...
0answers
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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 ...
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 $A$ ...
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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 ...
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 ...
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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 ...
0answers
39 views