Model-Free Closed-Loop Stability Analysis:
A Linear Functional Approach

Appendix 1: Mapping of the basis functions onto the unit disc

Applying transform (7) to the basis functions of the complex plane (3) yields the following:

\begin{align*} B_{k}^{\mathrm{disc}}\left(z\right) &=\sqrt{\pi\alpha}\frac{2}{z-1}B_{k}\left(\alpha\frac{1+z}{1-z}\right)\\ &=-\sqrt{\pi\alpha}\frac{2}{z-1}\sqrt{\frac{\alpha}{\pi}}\frac{\left(\alpha\frac{1+z}{1-z}-\alpha\right)^{k}}{\left(\alpha\frac{1+z}{1-z}+\alpha\right)^{k+1}}=z^{k} \end{align*}

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