Approximation and Decomposition of Functions in L^2 via Fractal Structures

This paper establishes a comprehensive theoretical framework for the approximation and decomposition of functions within the square-integrable space L^2([0,1]) by utilizing fractal structures. We focus on investigating the structural and approximation properties of functional spaces, denoted as V_n, which are generated through the translations and rescalings of a family of generating functions B over the elements of a fractal partition Gamma. Within this general setting, classical techniques such as Fourier series decomposition emerge naturally as special cases of our proposed decomposition scheme.

A central result of this study concerns the necessary conditions for universal approximation when the family consists of a single generator B. We prove a significant rigidity condition: the system constitutes a dense fractal system in L^2 if and only if B is constant almost everywhere. Furthermore, we derive explicit quantitative bounds for the speed of convergence when the approximation spaces are spanned by piecewise polynomials and trigonometric polynomials adapted to the fractal structure. By extending the classical theorems of Jackson and Bernstein–Walsh to the fractal domain, we demonstrate that the error decay depends on the regularity of the target function and the geometry of the structure. Specifically, for analytic functions, the error decays exponentially based on the scaling factor and structure level, while for differentiable functions, the error decays polynomially with respect to the degree and geometrically with the level. Finally, we discuss the representation of functions as series expansions and extend the fundamental density results to general metric spaces endowed with a finite Radon measure.