Steklov means and derivatives of positive order of Denjoy-integrable functions (Q1074801)

Essentially the author extends the results of \textit{P. L. Butzer, H. Dyckhoff, E. Görlich} and \textit{R. L. Stens} [Cand. J. Math. 29, 781-793 (1977; Zbl 0357.26003)] and \textit{P. L. Butzer} and \textit{D. Westphal} [Lect. Notes Math. 457, 116-145 (1975; Zbl 0307.26006)] employing finite Fourier transform method to the case of functions belonging to the space D of \(2\pi\)-periodic functions integrable in Denjoy Perron sense on the interval [0,2\(\pi\) ] with norm of \(f\in D^*\) given by \[ \| f\|_ D=\| f(\cdot)\|_ D=\sup_{v\in V_ 1}| \int^{2\pi}_{0}f(x)v(x)dx| \] where \(V_ 1\), is that class of \(2\pi\) periodic functions of bounded variation in [0,2\(\pi\) ] for which \(\sup_{0\leq x\leq 2\pi}| v(x)| \leq 1\) and \(_{0\leq x\leq 2\pi} v(x)\leq 1\). the Steklov means \(A_ h^{\alpha_ f}\) of \(f\in D^*\) of order \(\alpha\geq 1\) are defined by convolution of f and \(\chi\) (x;h), \[ \chi_{\alpha}(x;h)=2\pi \sum_{-x/2\pi <j<\infty}\frac{1}{h}p_{\alpha}(\frac{x+2\pi j}{h}),\quad h>0,\quad \alpha >0,\quad -\infty <x<\infty \] \[ and\quad p_{\alpha}(x)=\frac{1}{\sqrt{\alpha}}\sum_{0\leq j\leq x}(-1)^ j\left( \begin{matrix} \alpha \\ j\end{matrix} \right)(x-j)\quad^{\alpha - 1},\quad if\quad 0<x<\infty,\quad =0\quad if\quad -\infty <x\leq 0. \] The author proves the following approximate property of Steklov means: Theorem: If \(f\in D^*\) and \(\alpha\geq 1\) then \[ \| A_ h^{\alpha}f-f\|_ D\leq M_ 5(\alpha) w(h;f)_ D;\quad 0<h\leq 2\pi \] where \(w(\delta;f)_ D=\sup_{| t| \leq \delta}\| f(.+t)-f(.)\|_ D\quad (\delta \geq 0)\) and \(M_ 5(\alpha)\) is the suitable positive constant depending on \(\alpha\). Further the author investigates various conditions for the existence of the Liouville- Grunwald derivative using the above mentioned theorem. Lastly the author proves an approximation theorem for functions differentiable in the above mentioned sense.