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).