Long Range Dependence (LRD) in functional sequences is characterized in the spectral domain under suitable conditions. Particularly, multifractionally integrated functional autoregressive moving averages processes can be introduced in this framework. The convergence to zero in the Hilbert-Schmidt operator norm of the integrated bias of the periodogram operator is proved. Under a Gaussian scenario, a weak-consistent parametric estimator of the long-memory operator is then obtained by minimizing, in the norm of bounded linear operators, a divergence information functional loss. The results derived allow, in particular, to develop inference from the discrete sampling of the Gaussian solution to fractional and multifractional pseudodifferential models introduced in Anh et al. (Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459, 2016) and Kelbert (Adv Appl Probab 37(1):1–25, 2005).

功能序列中的长程依赖性（LRD）在适当条件下的光谱域中表征。特别是，可以在该框架中引入多分数集成函数自回归移动平均过程。证明了周期图算子的积分偏差在希尔伯特-施密特算子范数中收敛到零。在高斯场景下，然后通过在有界线性算子的范数中最小化散度信息功能损失来获得长记忆算子的弱一致参数估计量。得出的结果尤其允许从高斯解的离散采样到 Anh 等人中引入的分数和多分数伪微分模型进行推理。(Fract Calc Appl Anal 19(5):1161-1199, 2016; 19(6):1434–1459,