Skin lesion segmentation plays a crucial role in early diagnosis and treatment of skin diseases. While Convolutional Neural Networks (CNNs) excel at capturing local features, and Transformers are adept at modeling global dependencies, both struggle with extracting periodic features. The dermoscopic images of skin lesions often exhibit strong periodic textures in their details. To address these challenges, we propose FAN-UNet, a novel architecture that combines the strengths of Fourier Analysis Network (FAN)-based vision backbones with the U-Net framework. The core Vision-FAN layer integrates Fourier analysis with self-attention mechanisms, enabling the model to effectively capture both long-range dependencies and periodic relationships. Experimental results on multiple skin lesion segmentation datasets demonstrate that FAN-UNet achieves an optimal balance between model complexity and performance.