On some de la Vallée Poussin type discrete linear operators (Q1102477)

The authors define a de la Vallée-Poussin type trigonometric kernel and the associated trigonometric as well as algebraic approximating operator. Let t be a real variable and g a real or complex-valued \(2\pi\)-periodic continuous function of t. The positive integers j, m and the non-negative integers k, \(\ell\) are such that the number \(n=(1/2)(jm+km-k-1+\ell)\) is a non-negative integer. Further let \(t_{\upsilon}=(2\pi \upsilon /jm)\) \((\upsilon =0,\pm 1,\pm 2,...)\), \[ s_{jk\ell m}(t) =\frac{\sin (jmt/2)\sin \quad k(mt/2)\cos^{\ell}(t/2)}{jm^{k+1} \sin^{k+1}(t/2)},\text{ if } \sin (t/2)\neq 0, \] \[ s_{jk\ell m}(t) = 1\text{ if } \sin (t/2)=0, \] \[ S_{jk\ell m}(g,t)\quad =\sum^{jm- 1}_{\upsilon =0}g(t_{\upsilon})s_{jk\ell m}(t-t_{\upsilon}). \] The statements of theorems are too lengthy to be reproduced here. We state corollary to Theorem 2: \[ \| S_{221m}(g,t)-g(t)\| \leq (2/\sqrt{3}+4/\pi)\omega (g,\pi (n+1)), \] where \(\omega(g,\delta)\) denotes the modulus of continuity of g.