Variations around Jackson's quantum operator

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International Press of Boston
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Let 0<q<1, ω≥0, ω0:=ω/(1 − q), and I a set of real numbers. Consider the so-called quantum derivative operator, D_{q,ω}, acting on functions f:I→K (where K = R or C) as D_{q,ω}[f](x):=(f(qx+ω )−f(x))/((q−1)x+ω), x∈I\{ω0}, and D_{q,ω}[f](ω0):=f′(ω0) whenever ω0∈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,ω−integral, also called Jackson-Nörlund (q,ω)−integral. For ω=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,ω} and D_q, as well as between the q and the (q,ω) integrals, we show how to obtain the properties of D_{q,ω} and the (q,ω)−integral from the corresponding ones fulfilled by D_q and the q−integral. We also consider (q,ω)−analogues of the Lebesgue spaces, denoted by {\cal L}^p_{q,ω}[a, b] and L^p_{q,ω}[a, b], being a,b∈R. It is shown that the condition a≤ω0≤b ensures that these are indeed linear spaces over K. Moreover, endowed with an appropriate norm, L^p_{q,ω}[a,b] satisfies some expected properties: it is a Banach space if 1≤p≤∞, separable if 1≤p<∞, and reflexive if 1<p<∞.
Methods and Applications of Analysis