The local regularity of functional time series is studied under $L^p-m-$appro\-ximability assumptions. The sample paths are observed with error at possibly random design points. Non-asymptotic concentration bounds of the regularity estimators are derived. As an application, we build nonparametric mean and autocovariance functions estimators that adapt to the regularity and the design, which can be sparse or dense. We also derive the asymptotic normality of the mean estimator, which allows honest inference for irregular mean functions. Simulations and a real data application illustrate the performance of the new estimators.

翻译：在$L^p-m-$可逼近性假设下，本文研究了泛函时间序列的局部正则性。样本路径在随机设计点上含误差地被观测。我们推导了正则性估计量的非渐近集中界。作为应用，我们构建了能够适应正则性和设计（稀疏或密集）的非参数均值与自协方差函数估计量。此外，我们还推导了均值估计量的渐近正态性，从而实现对不规则均值函数的诚实推断。模拟实验与真实数据应用展示了新估计量的性能。