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.
From: Hailu Bikila Yadeta [view email]
[v1]
Mon, 3 Oct 2022 13:22:02 UTC (13 KB)
[v2]
Mon, 17 Jul 2023 06:13:30 UTC (12 KB)
[v3]
Mon, 14 Jul 2025 15:10:42 UTC (16 KB)
[v4]
Tue, 23 Jun 2026 13:00:46 UTC (15 KB)
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