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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 isThe 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.