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1answer
37 views

solution uniqueness of non-linear Fredholm equations

the equation is $F(x)=G(\int k(x,y)f(y)dy)$ $(*)$ where $f(x)=dF(x)/dx$ is the unknown and it's required to be non-negative. With integral by parts we'll have the form of a non-linear Fredholm ...
6
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0answers
159 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 ...
2
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0answers
91 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 ...
2
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0answers
110 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
votes
0answers
152 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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0answers
59 views

Convergence of solutions of the volterra integral equation with convergent kernels

Consider the following Volterra integral equation $$ g(t) = \int_0^t K_n(t,s)w_n(s) ds $$ where g(t) and K_n(t,s) are continuous and $K_n(t,s)\geq K_{n+1}(t,s)$ for all $t,s$. Moreover, $K_n(t,s)$ ...
1
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0answers
153 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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0answers
33 views

A source for integral operators in the context of Arthur-Selberg trace formula

Could you suggest a textbook for integral operators (in the context of Arthur-Selberg trace formula)?
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0answers
22 views

Uniqueness of homogeneous second kind Fredholm equation

I have the following equation: $-\frac{1}{2}\phi + W\phi - V(f\phi) = 0 $ and I intend to prove uniqueness of the solution in a appropriate Sobolev space. V and W denote the direct values of the ...
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0answers
48 views

volterra equation of the first kind with K(0,t)=0

A standard assumption both in theory and numerical methods for the Volterra equation of the first kind $$ g(t) = \int_0^t K(s,t) f(s) ds$$ is that $K(s,t) \neq 0$. One can show existence of the ...
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0answers
32 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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0answers
83 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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0answers
88 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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0answers
270 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: ...