= Solution
For one-column <Gram matrices>, unit <Euclidean norm> of the columns gives $A_S^*A_S=1$. The <restricted isometry constant> formula therefore gives $\delta_1=0$. For a two-column <Gram matrix>, put $c=\langle a_i,a_j\rangle$. The difference from the <identity matrix> has the form
$$
\begin{pmatrix}0&c\\\overline c&0\end{pmatrix},
$$
up to the convention for the complex <inner product>. Its characteristic polynomial is $\lambda^2-|c|^2$, so the <eigenvalues> are $\pm|c|$ and its <matrix 2-norm> is $|c|$. Maximizing over pairs gives $\delta_2=\mu$, where $\mu$ is the <mutual coherence>.
Part (b), together with the minimality defining the <restricted isometry constant>, gives $\delta_s\le\mu_1(s-1)$. Every summand defining <cumulative coherence> is at most the <mutual coherence>, so $\mu_1(s-1)\le(s-1)\mu$. \b[Thus, for normalized columns and $N\ge2$,]
$$
\boxed{\delta_1=0,\qquad\delta_2=\mu,\qquad\delta_s\le\mu_1(s-1)\le(s-1)\mu\quad(2\le s\le N).}
$$
The upper range $s\le N$ matters because the printed definition of <cumulative coherence> stops at $N-1$. For $N=1$, only $\delta_1=0$ is needed; the maximum over pairs defining <mutual coherence> is otherwise empty.
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