Exact Values of Kolmogorov Widths for the Classes of Analytic Functions. I

We prove that the kernels of analytic functions of the form

$$ {H}_{h,\beta }(t)={\displaystyle \sum_{k=1}^{\infty}\frac{1}{ \cosh kh} \cos \left(kt-\frac{\beta \pi }{2}\right),}h>0,\beta \in \mathbb{R}, $$

satisfy Kushpel’s condition C _y,2n starting from a certain number n _h explicitly expressed via the parameter h of smoothness of the kernel. As a result, for all n ≥ n _h , we establish lower bounds for the Kolmogorov widths d _2n in the space C of functional classes that can be represented in the form of convolutions of the kernel H _h,β with functions φ⊥1 from the unit ball in the space L _∞ .