It is well known that eigenfunctions of a kernel play a crucial role in kernel regression.Through several examples, we demonstrate that even with the same set of eigenfunctions, the order of these functions significantly impacts regression outcomes.Leveraging the Le Cam equivalence, a statistical equivalence between the sequence model and kernel regression, we introduce an over-parameterized gradient descent in the realm of sequence model to capture the effects of various orders of a fixed set of eigen-functions.This method is designed to explore the impact of varying eigenfunction orders.Our theoretical results show that the over-parameterization gradient flow can adapt to the underlying structure of the signal and significantly outperform the vanilla gradient flow method.Moreover, we also demonstrate that deeper over-parameterization can further enhance the generalization capability of the model.These results not only provide a new perspective on the benefits of over-parameterization and but also offer insights into the adaptivity and generalization potential of neural networks beyond the kernel regime.