Sharp Jackson inequality with a special modulus of continuity
Let \(\mathbb{T}=\mathbb{R}/\mathbb{Z}=[-1/2,1/2)\) and \(C(\mathbb{T})\) be the the set of continuous \(1\)-periodic functions. Let \(E_{n-1}(f)\) be the best approximation of a function \(f\in C(\mathbb{T})\) by trigonometric polynomials of degree at most \(n-1\). For \(h>0\) denote by \(\chi_h\), the characteristic function of the interval \((-h/2,h/2)\), multiplied by the factor \((1/h)\). Denote by \(\ast\) the convolution product. The authors consider the second order modulus: \(W_2(f,\chi_h)=\| f\ast\chi_h\|=\left\| h^{-1} \int_0^{h/2}[f(\cdot-t)-2f(\cdot)+f(\cdot+t)]dt\right\|\), \(f\in C(\mathbb{T})\), \(h>0\). The main result is: \[ E_{n-1}(f)\leq\mathcal{F}_n(\alpha) W_2(f,\chi_{\alpha/(2n)}),\text{ for } f\in C(\mathbb{T}),\; n\in\mathbb{N},\;\alpha>2/\pi, \] where \(\mathcal{F}_n(\alpha)=\sec(\alpha^{-1})+\tan(\alpha^{-1})\). Moreover, the constant is the best possible for \(\alpha=2\nu+1\), \(\nu=0,1,\dots\) In order to obtain this result, the authors develop a technique based on the expansion \(f=f-f\ast \chi_h+\chi_h\ast(f-f\ast\chi_h)+\chi_h^2\ast(f-f\ast\chi_h)+\dots\) The modulus \(W_2(f,\chi_h)\) is compared with the \(K\)-functional \(K_2(f,h)=\inf_{g\in C^2}\{\| f-g\|+h^2\| D^2g\|\}\) by the inequalities \((1/4)K_2(f,h/(4\sqrt{6}))\leq W_2(f,\chi_h)\leq 4K_2(f,h/(4\sqrt{6}))\).
scientific article; zbMATH DE number 6381062
scientific article