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Variations around a general quantum operator

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Abstract

Let \(\,I\subseteq {\mathbb {R}}\,\) be an interval and \(\,\beta :\,I\rightarrow \,I\,\) a strictly increasing and continuous function with a unique fixed point \(\,s_0\in I\,\) that satisfies \(\,(s_0-t)(\beta (t)-t)\ge 0\,\) for all \(\,t\in I\), where the equality holds only when \(\,t=s_0\). For appropriate choices of the function \(\,\beta ,\) the quantum operator defined by Hamza et al., \(\,D_{\beta }[f](t):=\displaystyle \frac{f\big (\beta (t)\big )-f(t)}{\beta (t)-t}\,\) if \(\,t\ne s_0\,\) and \(\,D_{\beta }[f](s_0):=f^{\prime }(s_0)\,\) if \(\,t=s_0,\) generalizes both the Jackson \(\,q\)-operator \(\,D_{q}\,\) and the Hahn (quantum derivative) operator, \(\,D_{q,\omega }\). With respect to the inverse of this general quantum difference operator, the \(\,\beta \)-integral, we study properties of the corresponding Lebesgue spaces \(\,{\mathscr {L}}_{\beta }^p([a,b]).\)

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Notes

  1. In fact, \({\mathbb {K}}={\mathbb {X}}\) can represent any Banach space [24, p.2]

  2. This shows that, for every fixed \(\,x\in \,I\), the sequence \(\{\beta ^{k}(x)\}_k\,\) is strictly monotone decreasing or strictly monotone increasing according to \(\,x>s_0\,\) or \(\,x<s_0\), respectively. Proposition 1 shows that it converges in both cases to the fix point \(\,s_0\).

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Acknowledgements

The author is grateful to J. Petronilho for the interesting discussions.

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  1. Departamento de Matemática da ECT, UTAD, Quinta de Prados, 5001-801, Vila Real, Portugal

    J. L. Cardoso

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This research was partially supported by Portuguese Funds through the FCT — Fundação para a Ciência e a Tecnologia—within the Project UID/MAT/00013/2013.

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Cardoso, J.L. Variations around a general quantum operator. Ramanujan J 54, 555–569 (2021). https://doi.org/10.1007/s11139-019-00210-8

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