In this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function $$\chi $$χ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function $$\sigma $$σ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, $$L^p$$L p -approximation is also established. At the end of the paper, examples of activation functions are discussed.

The analysis and processing of satellite remote sensing data have established new paradigms to observe the Earth and to study climate changes. Data collected by sensors and systems operating across the entire frequency spectrum carry on enormous information content. They enable systematic retrieval and characterization of bio-geophysical variables. The retrieval of these variables, however, is often hindered by limitations such as missing data, low spatial and temporal resolutions, and insufficient co-located ancillary data. These challenges arise due to factors such as sensor limitations, dead pixels, cloud cover, and discontinuous data acquisition. To overcome these issues, numerous techniques have been proposed, ranging from classical interpolation methods and spectral transforms to modern artificial intelligence-based approaches. This paper presents a comprehensive and critical review of satellite remote-sensing data augmentation methods, focusing on how interpolation theory and artificial intelligence techniques can be effectively applied to reconstruct missing data and enhance the quality of retrieved bio-geophysical variables. This roadmap offers a comprehensive framework that categorizes and evaluates existing methods, highlights their strengths and limitations, and identifies promising areas for future exploration.

Tangible climate changes are experienced by the Earth. They refer to long-term changes in, e.g., temperature and precipitation, as well as other aspects of the Earth system. Essential Climate Variables (ECVs) are used to monitor climate changes, validate climate models and, ultimately, inform policy decisions; they are, therefore, essential markers that help scientists understand the Earth system dynamics. Fundamental contributions to the understanding and characterization of ECV values can be given by the analysis of data produced by current and next-generation Earth observation (EO) satellites, which, among other options, can be based on microwave remote sensing. EO sensors can directly characterize bio-geophysical variables connected to ECVs, such as, e.g., the soil moisture (SM), the above-ground biomass (AGB), and the freeze-thaw (FT) dynamics. While FT dynamics are primarily relevant in northern latitudes, SM and AGB are globally significant parameters. SM, AGB, and FT are also key variables for the ESA EO science strategy and the ESA Climate Change Initiative program, mainly due to their strong relevance in understanding the global hydrological and carbon cycles. This work aims to be a comprehensive review of retrieval techniques for SM, AGB, and FT, considering data collected with passive and active microwave remote sensing systems. An overview of the methods proposed in the last decades is reported and discussed, then future perspective and current challenges are outlined. This paper targets a very broad scientific community, not necessarily engineers, but rather any scientist with interest in microwave EO and satellite remote sensing. For that reason, some basic concepts are briefly reported and discussed throughout the manuscript.

In this paper, we introduce a new family of neural network (NN) operators, namely the Steklov-type max-product NN operators. These operators extend the use of max-product techniques in function approximation by incorporating Steklov-type integrals. As it is well known, NN operators are typically generated by several kinds of sigmoidal activation functions, such as the ramp function, whose density functions can be expressed in terms of ReLU (Rectified Linear Unit). In particular, we establish pointwise and uniform convergence theorems for this class of operators in the space of continuous functions, and extend our analysis to the broader setting of Orlicz spaces, which include not necessarily continuous functions. The framework of Orlicz spaces further allows us to deduce convergence results in $$L^p$$ L p -spaces, $$1\le p<+\infty ,$$ 1 ≤ p < + ∞ , Zygmund spaces, exponential spaces and many others.