Best one-sided approximation of smooth multivariate functions (Q1915891)

Let \(L^q=L^q(T^d)\), \(1\leq q\leq\infty\), be the normed spaces on \(T^d=[-\pi,\pi]^d\) with usual norms. If \(W\subset L^q\) is a compact subset, then the best one-sided \(L^q\)-approximation of \(W\) by linear manifolds \(M\subset L^q\) of dimensions at most \(N\) can be characterized in terms of one-side \(N\)-widths \[ \widetilde d_N(W,L^q)=\inf_{\dim M\leq N} \sup_{f\in W} \inf_{u,v\in M} |u-v|, \] where \(u\) and \(v\) obey the condition \(u(x)\leq f(x)\leq v(x)\), \(x\in T^d\). For \(\alpha=(\alpha_1,\dots,\alpha_d)\), \(\alpha_j>0\), denote by \(\text{SH}^\alpha_p\), \(1\leq p\leq\infty\), the unit ball of the Hölder space \(H^\alpha_p\) of all functions \(f\) on \(T^d\), with zero mean value in each variable and finite seminorm \[ |f|_{H^\alpha_p}= \sup_{h\in T^d} |\Delta^2_h f^{(r)}_p\prod^d_{j=1} |h_j|^{-\beta_j}, \] where \(\Delta^2_h=\Delta_h\circ \Delta_h\), \(\Delta_h=\prod^d_{j=1} \Delta_{h_j}\), \((\Delta_{h_j}f)(x)=f(x_1,\dots,x_j+h_j,\dots,x_d)-f(x)\), \(f^{(r)}\) is the partial derivative of order \(r\) of \(f\); the vectors \(r\in Z^d_+\) and \(\beta\in (0,1]^d\) are uniquely determined from the equality \(\alpha=r+\beta\). The author finds (with some restrictions on \(\alpha\) and \(1\leq p<q\leq \infty)\) an upper estimate for \(\widetilde d_N(\text{SH}^\alpha_p,L^q)\). In the case \(q\leq2\) this estimate coincides with the asymptotic degree of the width \(\widetilde d_N(\text{SH}^\alpha_p,L^q)\).