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Questions tagged [integral-operators]

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17 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 ...
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1answer
48 views

Eigenvalues and eigenfunctions of an integral operator

Let $T$ be an integral operator with kernel $K(x,y)=|x-y|$ 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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0answers
9 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?
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1answer
35 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|...
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1answer
116 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
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1answer
25 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
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1answer
76 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 ...
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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 ...
3
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1answer
87 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
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1answer
48 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^...
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44 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 ...
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0answers
75 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 ...
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1answer
72 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 ...
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1answer
118 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} ...
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0answers
104 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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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 ...
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0answers
183 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
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1answer
119 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 ...
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147 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 ...
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0answers
122 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_\...
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1answer
36 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 \|...
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2answers
97 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 ...
3
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0answers
106 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
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1answer
830 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(...
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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
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1answer
202 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
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1answer
31 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
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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{...
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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,\...
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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 $L^{2p}$-...
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1answer
100 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 \...
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1answer
70 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)^{...
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1answer
170 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 ...
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1answer
246 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
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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 ...
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0answers
134 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 ...
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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, ...
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1answer
206 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
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1answer
350 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(...
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0answers
98 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 ...
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1answer
796 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 ...
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1answer
532 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([...
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0answers
93 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
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0answers
576 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 $\Omega \times \Omega$ ...
7
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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 ...
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0answers
318 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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1answer
100 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 ...
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1answer
159 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,\...
1
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0answers
111 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 ...
0
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1answer
36 views

Solving 2nd order linear ODE with integral transformation

I have this differential equation $-u''(x)+\mu \cdot u(x)=f(x)$ where $x \in (0,\pi)$ with boundary conditions $u'(0)=u'(\pi)=0$ where $c$ is a constant. I checked the values of $\mu$ where I have ...