Approximation of multidimensional functions with a given majorant of mixed moduli of continuity
Denote by \(\pi_d\) the cube \([-\pi,\pi]^d\), \(d\in \mathbb N\) and for \(q\geq 1\) let \(L_q^0(\pi_d)\) be the subspace of functions in \(L_q(\pi_d)\) for which \(\int_{-\pi}^{\pi}f(x) dx_i=0\), \(i=1,\dots,d.\) The classes \(H_q^\Omega\) of functions in \(L_q^0(\pi_d)\) are classes for which \(\Omega(f,t)_q=\sup\{\|\Delta_h^\ell f(x)\|_q\), \(|h_i|\leq t_i\), \(i=1,\dots,d\}\leq \Omega(t),\) where \(\Omega\) is a given function of the mixed \(\ell\)th-order modulus of continuity type. The author approximates the functions \(f\in H_q^\Omega\) (in the \(p\)-norm, \(p\leq q\)) by trigonometric polynomials of the form \[ T(x)=\sum_{k\in Q(N)}c_k e^{i(k,x)},\quad x=(x_1,\dots,x_q), \] where \(N\) is a given natural number and \(Q(N)=Q(N,\Omega)\) is a convenable set of multiindices generated by a level surface of the function \(\Omega.\) Under some conditions on \(\Omega\) one obtains the exact order for the upper bounds of the best approximation.
best approximation of functions by trigonometric polynomials
scientific article; zbMATH DE number 1407538
scientific article