Jackson-type theorems on a finite interval with weights having non-symmetric inner singularities (Q1307414)

The main result of this paper is the following. Theorem. If \(f\in C(w)\) then \(E_n(w;f)=O(\omega_\varphi(w f;n^{-\lambda}))\), where \(\{\alpha,\beta\}\subset (0,\infty)\), \(V_{\alpha,\beta}(x)=(-x)^\alpha\) if \(x\leq 0\), \(V_{\alpha,\beta}(x)=x^\beta\) if \(x>0\), \(\{s,n\}\subset\{1,2,\dots\}\), \(-1<t_1<\dots<t_s<1\), \(| x| \leq 1\), \(\{\alpha_i,\beta_i\}\subset(0,\infty)\) for every \(i\in\overline{1,s}\), \(w(x)=\prod^{i=s}_{i=1} V_{\alpha_i,\beta_i}(x-t_i)\) and \(C(w):=\{f:f\in C([-1,1]\setminus\bigcup^{i=s}_{i=1}\{t_i\}),\;\lim_{x\to t_i}(wf)(x)=0\;\text{for every} i\in\overline{1,s}\}\). Also \(\Pi_n\) is the set of polynomials of degree at most \(n\), \(\| \cdot \| \) is the supremum norm over the interval \([-1,1]\), for \(f\in C(w)\), \(E_n(w_i f)=\inf\{\| w(f-p)\| :p\in\Pi_n\}\), \(\gamma_i=\min\{\alpha_i,\beta_i\}\), \(\Gamma_i=\max\{\alpha_i,\beta_i\}\) for \(i\in\overline{1,s}\), \(\lambda=\min\{\min_{\Gamma_1\leq 1} \gamma_i, \min_{\Gamma_i>1}\gamma_i\Gamma^{-1}\}\) and \(\omega_\varphi\) is the Ditzian-Totik modulus of continuity.