2022/10/03 by Hailu Bikila Yadeta, Yadeta, Hailu Bikila
Mathematics · #Advanced Differential Equations and Dynamical Systems #Meromorphic and Entire Functions #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2210.00915
We prove that the space ℙp of p-periodic functions decomposes as the direct sum ℙp/2 ⊕ \mathbbAℙp/2, where ℙp/2 denotes the space of functions periodic with period p/2 and \mathbbAℙp/2 denotes the space of functions antiperiodic with antiperiod p/2 (i.e., f(x+p/2) = -f(x)). Iterating this decomposition yields a hierarchy of refined periodic subspaces. Under suitable uniform decay conditions on the residual periodic components, any p-periodic function on a compact interval admits a convergent expansion into a series of antiperiodic components with distinct antiperiods. As a concrete example, the continued periodic-antiperiodic decomposition of the fractional part function \x\ generates the Rademacher system. Additionally, we examine an orthogonal decomposition of L2(0,1) induced by reflection symmetry about the midpoint x = 1/2, i.e., f(x) = ± f(1-x). Using explicit projection operators, we show that this reflection-based decomposition generates a multiscale structure analogous to the Haar multiresolution analysis: the antiperiodic (odd-reflection) component yields a system equivalent to the Haar wavelet family \ψj,k\, while the periodic (even-reflection) component corresponds to the scaling space of piecewise constant functions. This provides a boundary-condition-based interpretation of the Haar wavelet basis.