## Taiwanese Journal of Mathematics

### DIFFERENTIATION ON VILENKIN GROUPS USING A MATRIX

Medo Pepić

#### Abstract

Given a Vilenkin group G, a scalar matrix $\Lambda = [\lambda_{ij}]_{i \in \mathbb{N}, j \in \mathbb{N}_0}$, a function $f \in L^1(G)$, and a point $x \in G$ we introduce, for each $\alpha \in \mathbb{R}$, the $(\Lambda,\alpha)$ − derivative $f$ at $x$ denoted by $f^{(\Lambda,\alpha)}(x)$. We also introduce the sets: $$M_\alpha = M(G,\Lambda,\alpha,x) := \big\{ f \in L^1(G)：\exists f^{(\Lambda,\alpha)}(x) \big\},$$ $$M = M(G,\Lambda,x) := \{f \in L^1(G)：\exists f^\Lambda(x)\};$$ where $f^\Lambda(x)$ derivative in [8], which is a generalization of Onneweer’s derivative $f^{[1]}(x)$ in [6]. We proved:

(a) Five theorems which express essential characteristics of $(\Lambda,\alpha)$− derivative,

(b) $M = M_0$,

(c) $(\forall \alpha,\beta \in \mathbb{R}) \wedge (\alpha \lt \beta) \Rightarrow (M_\alpha \subseteq M_\beta) \wedge (M_\beta \setminus M_\alpha \not= \theta)$.

Statement b) states that the method $(\Lambda,\alpha)$ − differentiation, for $\alpha = 0$, is equal to $\Lambda$ − differentiation and statement  c) says that $(\Lambda,\alpha)$− differentiation increases with increasing $\alpha \in \mathbb{R}$.

#### Article information

Source
Taiwanese J. Math., Volume 15, Number 6 (2011), 2387-2402.

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500406477

Digital Object Identifier
doi:10.11650/twjm/1500406477

Mathematical Reviews number (MathSciNet)
MR2896124

Zentralblatt MATH identifier
1253.43005

Subjects
Primary: 43A75: Analysis on specific compact groups

#### Citation

Pepić, Medo. DIFFERENTIATION ON VILENKIN GROUPS USING A MATRIX. Taiwanese J. Math. 15 (2011), no. 6, 2387--2402. doi:10.11650/twjm/1500406477. https://projecteuclid.org/euclid.twjm/1500406477

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