Yes, the analysis presented in this paper can be extended to higher-dimensional functions and more general noise models beyond the Gaussian assumption. The framework of minimax estimation is versatile and can be adapted to accommodate functions defined on higher-dimensional spaces, such as ( \\mathbb{R}^d ). In such cases, the smoothness properties of the function would need to be defined in a multidimensional context, potentially using concepts like ( \\beta )-Hölder continuity in higher dimensions.
Moreover, the analysis could be expanded to consider different types of noise models, such as heavy-tailed distributions or non-Gaussian noise, which are often encountered in real-world applications. By employing techniques from nonparametric statistics and robust estimation, researchers can derive new theoretical results that account for these complexities. This would not only enhance the understanding of the estimation problem in more general settings but also lead to the development of more robust and effective despeckling algorithms that can handle a wider variety of noise characteristics and dimensionalities.