We formulate Zeno and anti-Zeno dynamics in a finite-memory reservoir as a time-domain measurement-filter problem. A two-level system exchanges an excitation with a finite chain of damped memory modes and is subjected to repeated partial, nonselective measurements. In the weak-coupling limit, the population decay rate is written as an overlap between a finite reservoir memory kernel and a measurement-dependent filter. For any finite chain, this separation gives a compact expression that connects pulsed and continuous monitoring while retaining correlations across successive measurement intervals. When all memory-mode frequencies are tuned to the two-level transition, a single lossy mode gives monotonic Zeno suppression, whereas two modes form the smallest chain that can produce anti-Zeno-to-Zeno switching without an externally imposed onsite detuning. Exact stroboscopic Liouvillian calculations verify the analytic filter prediction and quantify finite-coupling corrections. For longer chains, the response is controlled by the frequencies, lifetimes, and boundary weights of the memory poles rather than by chain length alone.

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• Open quantum systems & decoherence
• Quantum Zeno dynamics
• Quantum measurements