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2
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0answers
47 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
31 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
47 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
44 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 $ ...
5
votes
1answer
113 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 $\Omega \times \Omega$ ...
0
votes
0answers
23 views

Solution of Volterra equation in $L_\infty$?

I'm having some trouble proving the following (if it is at all true?). Consider a time-varying Volterra equation $$ F(x, \xi) = f(x, \xi) + \int_\xi^x G(x, s) F(s, \xi) ds $$ for some (known) ...
3
votes
2answers
79 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 ...
0
votes
0answers
25 views

Neumann Series for integral equation with inhomogeneous term zero

Consider the method described in the following article: http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html In this notation, what happens when $ f(x)=0 $? All the terms seem to be zero ...
1
vote
0answers
49 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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
33 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
vote
0answers
17 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
votes
1answer
27 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 ...
1
vote
0answers
29 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^{\...
3
votes
1answer
80 views

Frechet Derivatives of a nonlinear integral operator

The nonlinear integral operator $P:C[0,1]\to C[0,1]$ is defined as follow: $$P(f)(x)=1+kxf(x)\int_0^1\frac{f(s)}{x+s}ds$$ In order to obtain the Frechet derivative of the operator, I start with: $$...
2
votes
1answer
83 views

Contraction operator

In a proof of Picard's theorem using the contraction mapping theorem, we define an operator $T$ which is applied to a function $y$. I don't really see below how $Ty$ is any different from $y$ as the ...
0
votes
0answers
46 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
vote
0answers
28 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 ...
5
votes
1answer
119 views

Proof of the integral operator in $L^2(\mathbb{R})$ being self-adjoint “by hand”

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(x) \, dy$$ This operator is bounded and $\|A\|=1$ (see Norm of the ...
4
votes
2answers
161 views

Norm of the integral operator in $L^2(\mathbb{R})$.

Suppose we have an integral operator $A$ such that $$Af(x) = \frac{1}{\sqrt{2\pi}}\int\limits_{\mathbb{R}}e^{-\frac{(x-y)^2}{2}}f(y)dy$$ To find $\|A\|$ we can use the unitary Fourier transform $F$, ...
0
votes
1answer
111 views

non compact closed range operator

Lately I've been reading Abramovich and Aliprantis' book 'An invitation to operator theory', chapter 2 (page 69) on bounded below operators. I would like to find an example of non-compact (and ...
2
votes
2answers
79 views

Operator on $L^2 (0,1)$ defined by convolution with $|x-y|^{-\alpha}$

Define $A: L^2 (0,1) \to L^2(0,1)$ $$Af(x) = \int_0^1 f(y) \frac{1}{|x-y|^\alpha} dy \quad , \quad \alpha \in (0,1)$$ For what values of $\alpha$ is it well defined? Bounded? Compact? I tried doing ...
1
vote
0answers
49 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
vote
0answers
46 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 $...
4
votes
1answer
123 views

Sufficient conditions on integration kernel for continuity of the integral operator

Suppose that we have a measure $d\mu(v)=e^{-|v|^2}dv$ on $\Bbb R^d$. We define a linear operator $$T[f](u)=\int_{\Bbb R^d} |u-v|^\beta f(v) d\mu(v).$$ I want to establish conditons on $\beta\in\Bbb R$...
6
votes
1answer
173 views

Several questions about integral operators.

I have been fumbling with expressions of the form \begin{equation} A\{f\}(s) = \int A(s,t)f(t)\operatorname{dt} \tag{$\star$} \end{equation} as a generalization of the matrix product. When looking ...
1
vote
0answers
56 views

Linear algebra references explaining matrix form of linear differential and integral operators

Some years ago I was in a lecture where I met for the first time the matrix representation of some differential and integral operators (once discretized). Back then, someone mentioned me that every ...
0
votes
1answer
159 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega t}e^{\...
0
votes
1answer
80 views

Norm of Integral Operator on $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$

There are similar question but the characterization of the space $E$ that I have gives me problem in computing the actual norm. Let $E=\{u \in C[0,1]\ ,\ u(0) = 0\}$ with the usual $\parallel \cdot\...
0
votes
1answer
20 views

Proving that $(Sf)(x)=1+\int_0^x t^2f(t)\,dt$ is $K$-Lipschitz

I want to show that $S:C[0,1]\to C[0,1]$ where $$(Sf)(x) = 1+\int_{0}^{x} t^2\cdot f(t) \,dt$$ is a Lipschitz map for $x\in[0,1]$. There I use the Euclidean norm on $\mathbb{R}$ and the uniform norm ...
1
vote
1answer
76 views

Does an integral operator with a symmetric integrable kernel have to be bounded on $L^2$?

Suppose $K(x,y)$ is a symmetric kernel. Let $\phi\in L^2(\Omega)$, where $\Omega$ everywhere is a domain in $R^n$. Can $\int_{\Omega}K(x,y)\,\phi(y)\,dy$ belong to $L^2$? In other words can an ...
1
vote
0answers
42 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(...
1
vote
1answer
181 views

Prove an integral operator is compact

The statement is like this, $K\subset\mathbb{R}$ is compact, the operator $A:L^\infty(K)\mapsto L^\infty(K)$ is defined by $f(x)\mapsto\int_K k(x,y)f(y)dy$. For $x\neq y$, $|k(x,y)|\leq M|x-y|^{\...
5
votes
1answer
168 views

Does a compact operator always have a kernel?

I am sorry if this question is stupid..... I raise it when I read Lax's book Functional Analysis. We know that some integral operators are compact, for example an integral operator from $L^2[Y]$ to $L^...
1
vote
2answers
240 views

What is the general form of linear operators on continuous functions?

I was wondering if there was a representation for a set of operators dense in the space of linear operators $B$ mapping $C(a,b) \to C(c,d)$. I thought that maybe integral operators give a general ...
2
votes
0answers
78 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
1answer
130 views

Defining the integral on an arbitrary metric space

I am trying to prove a version of Mercer's Theorem for an arbitrary compact metric space; that is, I do not wish to restrict myself to the space of real-valued continuous functions $C[a,b]$. I ...
2
votes
0answers
58 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 $k$ ...
1
vote
1answer
30 views

Bounded operator on continuous functions

Let $X=C([0,1])$ and $T: X \rightarrow X$ defined as $$(Tf)(t)=f(t)+f(0)$$ Prove $T$ is bounded. I was thinking about using the fundamental theorem of calculus in order to get some bounds on $f(0)$ ...
2
votes
1answer
107 views

Adjoint of an integral operator

I'm reading through a text about integral operators and I've come across the following theorem: Let $k:\mathbb{R}^2\rightarrow\mathbb{C}$ be a kernel, $T:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$...
3
votes
0answers
36 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
45 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-...
4
votes
2answers
241 views

Linear algebra : eigenvalues of an integral operator on polynomials

Consider the linear transformation $$ T : \left\{ \begin{array}{ccc} \mathbb{R}_n[X] & \to & \mathbb{R}_n[X] \\ P & \mapsto & \int_0^1 (X + t)^n\,P(t)\,dt \end{array}\right. $$ where ...
1
vote
1answer
104 views

Generalized functions as integral kernels on Hilbert spaces

I'm a physics student and I'm studying functional analysis. I've got a doubt about some operators defined by integral kernels that are generalized functions. Let $L_2(a,b)$ be the Hilbert space of ...
12
votes
1answer
289 views

Inequalities on kernels of compact operators

Suppose we have a $\sigma$-finite positive measure $\mu(v)$ on $\Bbb R^d$ and we have two positive kernels on $\Bbb R^d\times \Bbb R^d$ $k_1(v,u)>0$, $k_2(v,u)>0$. We define integral operators $$...