Sequential recommendation is crucial for personalized user experiences, but existing methods, especially Transformer-based models, face challenges with the quadratic computational complexity for long sequences and often lack mechanisms to explicitly capture diverse preference types, such as periodic patterns. In response to these constraints, a novel DualSpectrum Mamba based model for Sequential Recommendation (DSMRec) was proposed. DSMRec leverages the linear complexity of State Space Models (SSMs) via its core component, the Dual-Spectrum Mamba Block (DSMB). Within the DSMB, a Temporal Mamba Branch models sequential dynamics, while a novel Frequency Mamba Branch utilizes Fast Fourier Transform (FFT) and Mamba to capture frequency-domain characteristics representing stable and periodic user interests. These dual representations are adaptively merged by a Personalized Preference Fusion Layer. Systematic empirical investigations on four reference datasets confirm that DSMRec exhibits superior performance over state-of-the-art traditional, Transformer-based, and recent Mamba-based baselines, validating its effectiveness and efficiency.