The best algebraic approximation in Hölder norm. (Research announcement) (Q2721523)

Let \(\varphi(x):\sqrt{1-x^2}\), and for \(f\in C[-1,1]\) set NEWLINE\[NEWLINE\omega^r_\varphi(f;t):=\sup_{(h)<t}\sup_x|(\Delta^r_{\varphi(x)h}f)(x)|.NEWLINE\]NEWLINE Introduce the space \(\text{Lip}^\varphi_\alpha\) by the norm NEWLINE\[NEWLINE\|f\|_\alpha:=\|f\|_c+\sup_{t>0} \frac{\omega^r_\varphi(f;t)}{t^\alpha} ,\tag{1}NEWLINE\]NEWLINE and its closed subspace \(\text{lip}^\varphi_\alpha\) determined by the condition \(\lim_{t\to+0}t^{-\alpha}\omega^r_\varphi(f;t)=0\).NEWLINENEWLINENEWLINEThe authors announce the Bernstein-Jackson type results for functions of \(\text{lip}^\varphi_\alpha\), \(0<\alpha<1\), and approximation by algebraic polynomials in the norm (1). They also present an evaluation of \(K\)-functional of the couple \((\text{Lip}^\varphi_\alpha,W^r_\infty(\varphi^r))\), \(0<\alpha<1\).