Variations around Jackson’s quantum operator

Let $0 \lt q \lt 1$, $ \omega \geq0$, $ \omega_0:= \omega/(1-q)$, and $I$ a set of real numbers. Consider the so-called quantum derivative operator, $D_{q, \omega}$, acting on functions $f:I \rightarrow \mathbb{K}$ (where $ \mathbb{K}= \mathbb{R}$ or $ \mathbb{C}$) as\[D_{q, \omega}[f](x):= \frac{f \big(qx+ \omega \big)-f(x)}{(q-1)x+ \omega} \; , \quad x \in I \setminus \{ \omega_0 \} \; ,\]and $ \,D_{q, \omega}[f]( \omega_0):=f^ \prime( \omega_0) \,$ whenever $ \omega_0 \in I$ and this derivative exists. This operator was introduced by W. Hahn in 1949. Its inverse operator is given in terms of the so-called Jackson–Thomae $(q, \omega)$-integral, also called Jackson–Nörlund $(q, \omega)$-integral. For $ \omega=0$ one obtains the Jackson’s $q$-operator, $D_q$, whose inverse operator is given in terms of the so-called Jackson $q$-integral. In this paper we survey in an unified way most of the useful properties of the Jackson’s $q$-integral and then, by establishing links between $D_{q, \omega}$ and $D_q$, as well as between the $q$ and the $(q, \omega)$ integrals, we show how to obtain the properties of $D_{q, \omega}$ and the $(q, \omega)$-integral from the corresponding ones fulfilled by $D_{q}$ and the $q$-integral. We also consider $(q, \omega)$-analogues of the Lebesgue spaces, denoted by $ \mathscr{L}_{q, \omega}^p[a,b]$ and $L_{q, \omega}^p[a,b]$, being $a,b \in \mathbb{R}$. It is shown that the condition $a \leq \omega_0 \leq b$ ensures that these are indeed linear spaces over $ \mathbb{K}$. Moreover, endowed with an appropriate norm, $L_{q, \omega}^p[a,b]$ satisfies some expected properties: it is a Banach space if $1 \leq p \leq \infty$, separable if $1 \leq p \lt \infty$, and reflexive if $1 \lt p \lt \infty$.

Keywords

Jackson $q$-integral, Jackson–Nörlund $(q, \omega)$-integral, $(q, \omega)$-Lebesgue spaces, $q$-analogues

2010 Mathematics Subject Classification

33E20, 33E30, 40A05, 40A10

Full Text (PDF format)

Published 20 January 2016