Some estimates of the remainder in the expressions for the eigenvalue asymptotics of some singular integral operators. (English) Zbl 0948.45001
Some estimates of the remainder in known asymptotic formulas for convolution operator’s eigenvalues, where the kernels of the operators are \(|x-y|^{\alpha-1}\) and \(\log|x-y|\), are obtained using elementary methods. More precisely, the following formulas are proved:
\[
\lambda_n\biggl(\int_{-1}^1|x-y|^{\alpha-1} dy\biggr)=2\Gamma(\alpha)\left(\dfrac 2{n\pi}\right)^{\alpha}\cos\dfrac{\alpha\pi}2 (1+O(n^{-\delta})),
\]
where \(0<\alpha<1\) and \(\delta\) depends on \(\alpha\) (the formula is explicitly given);
\[
\lambda_n\biggl(\int_{-1}^1-\dfrac 1{\pi}\log|x-y|dy\biggr)=\dfrac 2{n\pi}\biggl(1+O\biggl(\dfrac{\sqrt{\log n}}{n^{1/5}}\biggr)\biggr)
\]
and \(\lambda_n(H)=\frac{\pi n}2(1+O(n^{-1/11}))\), where
\[
Hf(x)=\dfrac 1\pi\int_{-1}^1\dfrac{f'(t)}{x-t} dt
\]
with \(f\in\operatorname{Dom}H\) iff \(f'\in L^1(-1,1)\), \(Hf\in L^2(-1,1)\), \(f(-1)=f(1)=0\) (the integral is taken in the sense of principal value).
Reviewer: Zoran Kadelburg (Zemun)
MSC:
45C05 | Eigenvalue problems for integral equations |
47G10 | Integral operators |
45P05 | Integral operators |