On a new mean and functions of bounded deviation
Let \(L_{2\pi}\) be the space of all \(2\pi\)-periodic and Lebesgue integrable functions over \([0,2\pi]\). Then for \(f\in L_{2\pi}\) and real \(x\), write \[ \begin{aligned} \varphi(t) &= \tfrac 12 \{f(x+t)+ f(x-t)\}= \varphi_0(t),\\ \varphi_\alpha(t) &= \alpha t^{-\alpha} \int_0^t (t-u)^{\alpha-1} \varphi(u) du \quad(\alpha>0),\\ P(t) &= \varphi(t)- \varphi_1(t). \end{aligned} \] Chandra's mean \(P(t)\), as pronounced by the authors, was first studied by the present reviewer [Period. Math. Hung. 5, 121-130 (1974; Zbl 0286.42010)]. In the present paper, generalizing this mean in the form: \[ \begin{aligned} p(k,t) &= \sum_{m=0}^k (-1)^m \binom km \varphi_m(t) \quad (k=1,2,\dots),\\ p(0,t) &= \varphi(t), \end{aligned} \] the authors have proved the following: Theorem 1. For each positive integer \(k\) \[ \begin{aligned} \varphi(t)\in \text{BV}(0,\pi) &\Leftrightarrow p(k,t)\in \text{BV} (0,\pi) \quad\text{and}\quad \frac{p(k,t)}{t}\in L(0,\pi), \tag{a}\\ p(k,t)\in \text{BV}(0,\pi) &\Rightarrow p(k+1,t)\in \text{BV} (0,\pi), \tag{b}\\ t^{-1} p(k,t)\in L(0,\pi)&\Rightarrow t^{-1} p(k+1,t)\in L(0,\pi). \tag{c}\end{aligned} \] The case \(k=1\) of part (a) was earlier known to the present reviewer [Jñānābha, Sect. A 8, 17-24 (1978; Zbl 0478.26003)]. In Theorem 2, the authors have demonstrated, by using Chandra's mean, that the class of functions of bounded deviation (BD) strictly includes the class of functions of bounded variation (BV).
scientific article; zbMATH DE number 1348067