= Solution
The projected actuator pitch is $p=D/N$ in the question's sampling convention. A sinusoidal <wavefront error> needs at least two samples per spatial period, by the <Nyquist–Shannon sampling theorem>. Thus the shortest limiting correctable scale at the primary is
$$
\boxed{\Lambda_{\min}=2p=\frac{2D}N.}
$$
The <Nyquist spatial frequency> is $N/(2D)$ cycles per unit length. Exactly at that limiting frequency some phases are poorly sampled; actual <deformable mirror> performance also depends on actuator influence functions, so this is an ideal bandwidth limit.
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