MRMFunktion

MRMFunktion is a theoretical mathematical function used to model data across multiple resolutions. In its typical formulation, the function takes an input x from a domain X and produces an output y in Y by combining information gathered at several scales.

Formally, MRMFunktion can be expressed as f(x) = sum_{l=0}^L f_l(x), where each f_l represents the contribution from resolution level l, computed through localized basis functions phi_{l,i} and corresponding weights w_{l,i}. The construction emphasizes locality and hierarchical structure, so that changing x at fine scales influences higher-level outputs in a controlled manner. Depending on the implementation, the function may be linear in x or nonlinear.

Construction and learning: The level-wise components f_l are derived from a partition of the input domain into regions or blocks. Basis functions are associated with regions, and coefficients are learned from data through optimization or predefined rules. The final output combines these level-wise terms, often with normalization to balance scales.

Applications: MRMFunktion appears in theoretical analyses of multi-scale data representations, in signal and image processing, time-series analysis, and feature extraction for machine learning. It is used to capture both global structure and local variations.

Relation and further reading: MRMFunktion relates to multi-resolution analysis, wavelet frameworks, and hierarchical feature maps. It is described in theoretical discussions and in some educational materials as a conceptual tool for understanding scale-based representations.