Decomposition of Spaces of Periodic Functions into Subspaces of Periodic and Antiperiodic Functions and Its Connection to the Rademacher System and the Haar Wavelet Basis

Abstract

We prove that the space $\mathbb{P}_p$ of $p$-periodic functions decomposes as the direct sum $\mathbb{P}_{p/2} \oplus \mathbb{A}\mathbb{P}_{p/2}$, where $\mathbb{P}_{p/2}$ denotes the space of functions periodic with period $p/2$ and $\mathbb{A}\mathbb{P}_{p/2}$ 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 $L^2(0,1)$ induced by reflection symmetry about the midpoint $x = 1/2$, i.e., $f(x) = \pm f(1-x)$. 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 $\{\psi_{j,k}\}$, 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.