A note on the rate of convergence of Schauder decomposition (Q1820387)

The paper contains a vast improvement of the author's results [(*) ibid. 102, 203-210 (1984; Zbl 0544.46008)] establishing a new rate of convergence of the sequence \(\{Q^ n_ k\}^{\infty}_{k=1}\) being a Schauder decomposition of the Sobolev space \(W_ p^{(n)}\). In Numer. Funct. Anal. Optimization 4, 45-59 (1981; Zbl 0483.46013) the explicit form of \(Q^ n_ k\) was derived and in (*) the author has proved that \(\| Q^ n_ kf-f\|_{\infty}=0(\omega (f^{(n)};1/3^ k))\) where \(f\in W_ p^{(n)}\cap C^ n[0,1]\), \(p>1\), \(n\geq 1\), \(\omega\) (f;n) denotes the modulus of continuity of f. Now he proves the estimations (Theorems 1 and 2): \[ \| Q^ n_ kf- f\|_{\infty}\leq cons\tan t\quad (1/3^ k)^{n-1}\cdot \omega (f^{(n-1)};1/3^ k)\quad for\quad f\in W_ p^{(n)}, \] \[ \| Q^ n_ kf-f\|_{\infty}\leq cons\tan t\quad (1/3^ k)^ n\quad for\quad f\in C^ n[0,1], \] \[ \| (Q^ n_ kf- f)^{(j)}\|_{\infty}\leq cons\tan t\quad (1/3^ k)^{k-1-j}\cdot \omega (f^{(n-1-j)};1/3^ k)\quad for\quad j=0,1,...,n-1;\quad f\in W_ p^{(n)}. \] In addition (Theorem 3) a sufficient condition is proved for uniqueness of the solution of Fredholm integral equation \(x_ x-Q^ n_ kKx_ k=Q^ n_ k,\) where \(Kx(t)=\int^{1}_{0}k(t,s)x(s)ds\) and a rate of convergence of \(x_ k\) to the solution x of the equation \(x-Kx=f\).