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
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 $\{ψ_{j,k}\}$, 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.