Questions tagged [integral-operators]
The integral-operators tag has no usage guidance.
0
votes
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 ...
4
votes
0answers
52 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 ...
0
votes
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
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 ...
4
votes
1answer
75 views
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?
0
votes
1answer
56 views
Operator norm of integral operator
Suppose we have $X=L^2([0,1];\mathbb{R})$ and
\begin{equation}
T:X\rightarrow X, \ Tf(x)=\int_0^1x^2yf(y)dy.
\end{equation}
Show that $T$ is compact and determine $||T||.$
I already have that $||T|...
1
vote
1answer
129 views
Find point spectrum and spectrum of integral operator
Let $A:L^2(0,\pi) \to L^2(0,\pi)$ be defined by $(Af)(x)=\displaystyle\int_{0}^\pi \sin(x-y)f(y)dy$. Find the point spectrum and spectrum of $A$.
I am not sure how to go about this. I thought to ...
1
vote
1answer
28 views
Kernel decomposition of a finite rank integral opeartor
Given a self-adjoint finite rank integral operator P on $L_2[0,1]$, it has the eigen-decomposition $P=\sum_{i=1}^k\lambda_i \langle u_i,\cdot\rangle u_i$ where $u_i$ are eigenfunctions and $\lambda_i$ ...
1
vote
1answer
104 views
Find eigenvalues and eigenfunctions for integral operator
I'm trying to find the eigenvalues and eigenfunctions for the integral operator
$Ku=\displaystyle \int_{-1}^1 (1-|x-y|) \,u(y) \, dy$
Since I want to find $\mu,u$ such that $Ku=\mu u$, we get the ...
1
vote
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 ...
3
votes
1answer
90 views
Is this operator on $L^\infty$ injective / surjective?
$$
f \in L^\infty (0,1) \\
Tf(x) = \int_0^x e^{y-x}f(y)dy, x\ge0
$$
I've shown that T is a bounded linear operator from $L^\infty(0,\infty)$ into itself. I've computed its norm (it should be $\|T\| = ...
3
votes
1answer
51 views
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^...
0
votes
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 ...
1
vote
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 ...
0
votes
1answer
74 views
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 ...
-1
votes
1answer
120 views
Finding the norm of an integral operator.
Let $H = L_{2}([-\pi,\pi])$. For each $m \in \mathbb{N^{*}}$ and define $T_{m}:H \rightarrow H$ by $$T_{m}(f)(t) = \frac{\int_{-\pi}^{\pi} f(s) \sin(2m (s+t)) ds}{2^m},$$
Let $T = \sum_{m=1}^{\infty} ...
4
votes
0answers
107 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 $A$ ...
6
votes
2answers
181 views
How can I show that this operator is bounded on $L^2$?
Consider the integral operator
$$Tf(x) = {\int}_{-\infty}^{\infty} \frac{\sin(x - y)}{x - y}f(y)dy$$
How can I show that $T$ is bounded on $L^2$?
I know that bounded means there is a ...
1
vote
0answers
190 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
$$Kf(y)=\int_{0}^1 \sin(\pi(x-y))f(y)...
3
votes
1answer
127 views
An explicit example of a compact integral operator in one dimension to help with intuition?
I am about to start studying compact operators and I always find the best way to get intuition on a new area is to start with a simple example. So what would be a an example of a compact integral ...
1
vote
0answers
150 views
Diagonal of kernel
Suppose that $K:L^2([0,1],\mathbb C)\to L^2([0,1],\mathbb C)$ is an integral operator given by
$$
Kf(x)=\int_0^1k(x,y)f(y)dy
$$
for each $f\in L^2([0,1],\mathbb C)$ and $x\in[0,1]$. In general, the ...
3
votes
0answers
142 views
Singular value decomposition of a specific integraloperator
I want to determin the singular value decomposition of the integral operator
$$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_\...
0
votes
1answer
37 views
Relation of a specific integral operator to the Laplacian
I am trying to make sense of the following operator, acting un the ser of continuos functions from $\mathbb{R}^2$ to $\mathbb{R}$:
$$\mathbf{L}[u]=lim_{r\to0}\frac{1}{r^2}(\frac{1}{r^2}\oint_{\left \|...
1
vote
2answers
98 views
Is $\sum_{k=-\infty}^{+\infty}\dfrac{\cos((4k+1)x)}{(4k+1)^n}$ a polynomial in $x$ for $n\in \mathbb{N}$?
My guess is that
$$\sum_{k=-\infty}^{+\infty}\dfrac{\cos((4k+1)x)}{(4k+1)^n}$$ might be a polynomial on $(0,\pi/2)$ with real coefficients in $x$ for $n\in \mathbb{N}$.
Since $\cos(0)=1$ the constant ...
4
votes
0answers
110 views
Is this integral operator on $L_2(0, \infty)$ compact?
Let's define $T:L_2(0,\infty) \to L_2(0,\infty)$ as $$(Tf)(x) = \int_0^\infty \frac{f(y)\sqrt{xy}}{x^2y^2+1}dy.$$ I'm interested, if this operator is compact.
$T$ is integral operator with kernel $K = ...
2
votes
1answer
853 views
Meaning of the inverse of a differential operator
Consider the Poisson's equation $$\nabla^2\phi(\textbf{x})=-\rho(\textbf{x})/\epsilon_0.$$ What is the meaning of the inverse operator in the following $$\phi(\textbf{x})=-\frac{1}{\nabla^2}\frac{\rho(...
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 ...
11
votes
1answer
205 views
Identity operator on $L^2(\mathbb{R}^d)$
I want to show that the identity operator on $L^2(\mathbb{R}^d)$ cannot be given by an absolutely convergent integral operator. That is, if $K(x,y)$ is a measurable function on $\mathbb{R}^d \times \...
0
votes
1answer
33 views
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 $C^1([-1,1])^*$...
2
votes
1answer
92 views
Difficulty proving continuity of integral operator.
I am trying to prove that given
$$g:I^{n+1}\subset \mathbb{R}^{n+1}\rightarrow \mathbb{R}^n$$
continuous, $(x_0,y_0)\in I^{n+1}$ and a sequence of functions defined iteratively as for each $f:\mathbb{...
1
vote
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,\...
0
votes
1answer
34 views
$\alpha \in L(L^p, L^{2p})\; \Rightarrow\; \alpha \in L(L^{2p}, L^{4p}), \, p \in [1,\infty)?$
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 $L^{2p}$-...
3
votes
1answer
106 views
Integral Operator Satisfying Holmgren Condition is Bounded
Consider the integral operator
$$u(x) = kf(x) = \int_{-\infty}^\infty k(x,s)f(s)ds.$$
Assuming the kernel $k(x,s)$ satisfies the Holmgren condition:
$$ \sup_{y \in \mathbb{R}} \int_{-\infty}^\infty \...
0
votes
1answer
73 views
Are these mappings linear and continuous functionals? Riesz representation theorem.
Which of the mappings $x\rightarrow F(x)$ are linear and continuous functionals over $L^2(0,1)$? .
$$(i)\qquad F(x)=\int_0^1\frac{x(t)}{\sqrt{t}}dt$$
$$(ii)\qquad F(x)=\left(\int_0^1x(t)^2dt\right)^{...
2
votes
1answer
177 views
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 ...
1
vote
1answer
256 views
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 ...
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 ...
1
vote
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 ...
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
$$
Af(y) = \int_0^1 k(x,y) f(x) dx, ...
1
vote
1answer
208 views
Eigenvalues and Eigenfunctions of a Particular Self Adjoint Operator.
Consider the operator $T(f)(x): L^2[0,1] \rightarrow L^2[0,1]$ defined $$T(f)(x)=\int_{0}^{1-x} f(y) \ (1-y-x) \ dy.$$ (Assume $L^2[0,1]$ is the set of square integrable real valued functions over the ...
3
votes
1answer
365 views
Methods for finding decompositions of Hilbert-Schmidt integral operators
For a Hilbert-Schmidt integral operator
$$(Kf)(x) = \int_Y k(x,y)f(y) dy$$
a decomposition (called Hilbert-Schmidt decomposition) of the following form exists:
$$k(x,y) = \sum_n \sigma_n u_n(x)v_n(...
2
votes
0answers
99 views
Let $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)\,\mathrm{d}t}$ (the Hardy operator) find the norm of $T$ on $L^p$ [duplicate]
We have the operator $T: L^p(\mathbb{R}^+) \to L^p(\mathbb{R}^+) $ with $p \in (1,+\infty)$, defined by $T(f):=\frac{1}{x}\int_{0}^{x}{f(t)dt}$.
We define $\tilde{f}(x)=e^{x/p}f(e^x)$ for all $f \in ...
1
vote
1answer
842 views
How to show that this integral operator is bounded?
Consider the integral operator $T : C([0,1])\to C([0,1])$ given by
$$Tf(t)=\int_0^1 K(t,\tau)f(\tau)d\tau.$$
I'm solving one exercise which is to show this operator is bounded. The exercise is from ...
1
vote
1answer
544 views
An expression for the Hilbert-Schmidt inner product
Suppose that $k:[0,1]\times[0,1]\to\mathbb C$ is a Hilbert-Schmidt kernel, i.e.
$$
\int_0^1\int_0^1|k(x,y)|^2\mathrm dx\mathrm dy<\infty.
$$
The associated Hilbert-Schmidt integral operator $K:L^2([...
0
votes
0answers
94 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 $ ...
7
votes
0answers
614 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 $\Omega \times \Omega$ ...
8
votes
2answers
1k views
Intuition behind: Integral operator as generalization of matrix multiplication
So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications.
In ...
1
vote
0answers
327 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
vote
1answer
103 views
Regarding integral operators being contractions
I have two half-questions that tie into one another.
Suppose $T$ is an operator on $C([0, 1])$ defined by $$(Tu)(t) = \int_0^t (u(x))^2dx.$$
Show that T is not a contraction on the closed unit ball ...
1
vote
1answer
160 views
About the spectral radius of an integral operator
My question is given at the end of the explanation.
Let $K\in{}C([a,b]^{2},\mathbb{R})$ and consider the operator $H:C([a,b],\mathbb{R})\to{}C([a,b],\mathbb{R})$ defined by
$$H[x](t):=\int_{a}^{t}K(t,\...
