On the Construction of Schauder Bases in Hilbert Spaces via Unitary Representations

Abstract

This article develops a unified framework for constructing Schauder bases in Hilbert spaces from unitary representations of locally compact groups, with emphasis on the affine action (the ax+b group) and its wavelet realization. We begin with the contrast between Hamel bases (algebraic existence) and Schauder bases (topological reconstruction), and show how topology—via continuity of coordinate functionals and convergence in norm—guides the validity of expansions useful in functional analysis. At an abstract level, we review Haar measure, regular representations, and the notion of a cyclic vector, and we state Schauder-type criteria for systems generated by orbits \(\{\pi(g)f\}_{g\in G}\). For the affine group, we recall the continuous wavelet transform, admissibility, and the reproduction formula; we then discretize on a dyadic lattice to obtain orthonormal (hence Schauder) systems in \(L^2(\mathbb{R})\) via multiresolution and quadrature mirror filter (QMF) conditions. The Haar wavelet appears as a prototypical case: its discrete orbit under dilations and translations generates a complete orthonormal basis. On the computational side, we implement simulations comparing Haar approximations with Fourier series on \([-3,3]\). We consider three representative functions: \(t^2\) (nonperiodic), rectangular wave with \(T=1\), and triangular wave with \(T=1\). We show that, for periodic functions, the Fourier series must be computed with the natural period (an indispensable correction), and that Haar offers localization advantages and robustness near discontinuities (mitigating Gibbs phenomena). For nonperiodic functions, the implicit periodization in Fourier introduces global artifacts that Haar partially avoids. We conclude by pointing to two directions for extension: (i) more regular wavelets (Daubechies, Riesz bases) and extensions to Banach spaces via coorbit theory and its discretization; and (ii) more general group actions (e.g., anisotropic semidirect products) tailored to specific geometries. The results strengthen the bridge between algebraic generation by group actions and stable reconstruction in functional analysis.