The integral-operators tag has no wiki summary.

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478 views

### Decomposition of an integral operator into a composition

I've been musing about the following question for a while now. Given an integral operator $G$ defined by
$$ (Gf)(x) = \int_0^1 G(x,u) f(u)\,du $$
Is it possible to decompose this into two separate ...

**6**

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**0**answers

151 views

### Density of odd and even eigenstates of an integral operator

Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...

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77 views

### Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as
$
\hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx.
$
The Hilbert transform ...

**2**

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**0**answers

119 views

### Schwartz kernel of a pseudodifferential operator with singular symbols

If we have a smooth symbol $r(x,\xi)$ of order $d$ the corresponding pseudodifferential operator $P$ is integral operator with kernel given by
$$K(x,y)=\int_{\mathbb{R}^n}e^{i(x-y)\cdot \xi} ...

**1**

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74 views

### Estimating singular values of integral operators

I would like to estimate the singular values of certain trace class integral operators. For the sake of concreteness, consider on $L^2({\mathbb R},dx)$ the integral operator $$(Tf)(x)=\int_{\mathbb ...

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116 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty ...

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23 views

### Multiple integral of the resolvent kernel

I am trying to implent Volterra equations using resolvent kernel.To do this, the iterative kernel
$$K_i(x,y) = \int\limits_x^y K_1(y,t)K_{i-1}(t, x)dt. $$
should be calculated. However, it is not ...

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53 views

### Existence and Uniqueness of Volterra integral equations of the first kind

$$
\int_0^t k(s,t)f(s)ds=g(t)
$$
To prove the existence and uniqueness of solutions of Volterra integral equation(VIE) of the first kind, we usually differentiate it and convert to the VIE of the ...

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67 views

### can we generalize the cauchy principal value and hadamard's finite part for multiple integrals $ n>1 $

I have seen in wikipedia, that how the Cauchy's pricnipal value and Hadamard's finite part work in dimension one
My question is when can we (or if negative answer why can not ) generalize the ...

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65 views

### Fredholm Integral Involving Stochastic Process

I wish to solve an integral equation of the form $$g(X) = c\int_0^1 K(X,t)f(t) \ dt $$ where $f\in L^1([0,1])$ and $g$ is some function on finite sequences of random variables. So, $X$ is a stochastic ...

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77 views

### $L^p$-derivative of Log

Let $\gamma$ be a smooth closed planar curve. Is $\int_\gamma\big(\frac{\log|\zeta-z|-\log|\zeta-w|}{|z-w|}\big)^2ds(\zeta)$ uniformly bounded in $z,w$ as $|z-w|$ is sufficiently small?another word ...

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43 views

### How to solve a multidimensional integral equation of convolution type 2 with real coefficients?

Can someone suggest suitable reference for solving a 4*4 integral equation of convolution, and whether the following equation has a closed form solution? I really appreciate your help.
where y, K, ...

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220 views

### Notation for a functional L2 matrix norm

Hi,
Let $v(z)$ be a 2x2 matrix depending on a complex variable z, defined on an oriented contour $\Sigma$ in the complex plane. Can anyone tell me the meaning of the notation:
...