Modern weather stations in Germany record daily temperatures every 10 minutes, whereas measurements from historical reference periods are often only available at much coarser temporal resolutions, typically hourly. This discrepancy must be accounted for when comparing historical and current daily temperature patterns. Motivated by this problem, we develop two-sample inference procedures for functional data under sampling schemes where one sample is densely observed while the other is relatively sparse. Building on recent ideas from transfer learning for functional data, we derive estimators of the difference of the mean functions that attain optimal convergence rates in the supremum norm. We further establish a functional central limit theorem in the space of continuous functions and develop multiplier bootstrap methods for constructing uniform confidence bands. Extensions to functional time series are also discussed. Applying the proposed methodology to daily temperature curves from German weather stations, analyzed separately by month, reveals that climate change has altered not only average temperatures but also intraday temperature patterns. In particular, for stations such as Berlin, warming from morning to early afternoon exceeds the daily average increase, whereas evening and nighttime temperatures exhibit comparatively smaller increases.

翻译：德国现代气象站每10分钟记录一次每日气温，而历史参考期的测量通常仅在更粗糙的时间分辨率下可用，通常为每小时一次。在比较历史和当前的每日温度模式时，必须考虑这种差异。受此问题启发，我们针对一个样本密集观测而另一个样本相对稀疏的采样方案，开发了函数数据的双样本推断程序。基于函数数据迁移学习的最新思想，我们推导了均值函数差异的估计量，并使其在 supremum 范数下达到最优收敛速度。我们进一步在连续函数空间中建立了函数中心极限定理，并开发了乘子自举方法以构建均匀置信带。还讨论了在函数时间序列上的扩展。将该方法应用于德国气象站的每日温度曲线（按月分别分析），揭示了气候变化不仅改变了平均气温，还改变了日内温度模式。特别是对于柏林等站点，从早晨到下午的变暖幅度超过了每日平均增加量，而傍晚和夜间的温度增加相对较小。