PomeDefinition 22.1.86.54C62141C0D9 Let $N\in \mathbb {N}$, and $A_{0},\ldots ,A_ N$ be a finite collection of elements in $\mathrm{PSH}(X,\theta )$, and $\tau _0>\tau _1>\cdots >\tau _ N$ be finitely many real numbers. Then the piecewise linear curve $A=(A_{\tau })_{\tau \in \mathbb {R}}$ in $\mathrm{PSH}(X,\theta )\cup \{ -\infty \} $ associated with these data is the affine interpolation of these data:
FE3BE40BF529$A_{\tau _ i}=A_ i$ for $i=0,\ldots ,N$;
2D3D174A0348$A_{\tau }=A_{\tau _ N}$ for $\tau \leq \tau _ N$;
B0B44E5310EAfor $t\in (0,1)$ and $i=0,\ldots ,N-1$, we have
\[ A_{(1-t)\tau _ i+t\tau _{i+1}}=(1-t)A_{\tau _ i}+tA_{\tau _{i+1}}; \]8FF2573D4019$A_{\tau }\equiv -\infty $ for $\tau >\tau _0$.
The analytification of $A$ is the function $A^{\mathrm{An}}\colon X^{\mathrm{An}}\rightarrow [-\infty ,\infty )$ defined as follows:
2B233D0F0F88
22.1.87
\begin{equation} A^{\mathrm{An}}(v)\coloneqq \sup _{\tau \leq \tau _0} (A_\tau ^{\mathrm{An}}(v)+\tau )=\max _{i=0,\ldots ,N} \left(A_{\tau _ i}^{\mathrm{An}}(v)+\tau _ i\right)\quad \forall v\in X^{\mathrm{An}}. \end{equation}
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