Estimates of the widths of classes of analytic functions (Q808360)

Let \(\beta \in R\), \(1<p,q<\infty\) and let K be an integrable function on (-\(\pi\),\(\pi\)) with the Fourier series \(K(\tau)\sim \sum^{\infty}_{k=1}\psi (k)\cos (k\tau -\beta \pi 2).\) Consider the class of convolutions \(N=\{K\cdot \phi +C: \| \phi \|_ q\leq 1\}\) where \(C\in R\) is a fixed constant, and by \(d_ n(\psi,\beta,p,q)\) denote the Kolmogorov n-width of the set N in \(L_ q(-\pi,\pi)\). The author proves that if \(\psi (k)=\exp (-\alpha k^ r)\) with \(\alpha >0\) and \(r\geq 1\), then there exist constants \(c_ 1(p,q)\), \(c_ 2(p,q)\) such that \[ c_ 1(p,q)\exp (-\alpha n^ r)\leq d_{2n}(\psi,\beta,p,q)\leq d_{2n-1}(\psi,\beta,p,q)\leq c_ 2(p,q)\exp (-\alpha n^ r). \]