Generalized Harmonic Decomposition Theorem for Non-Periodic Sequences

We give a concise, self-contained presentation of the Bohr–Besicovitch Fourier theory fordiscrete sequences in the Besicovitch space B2(Z). The emphasis is on the correct hypothesesand statements for sequences that are Besicovitch almost periodic (BAP): convergence is inthe B2 seminorm, coefficients are defined by Besicovitch means, and uniqueness holds in B2.We include a clean characterization of BAP via pure point spectral measures, arithmeticexamples (periodic classes and Dirichlet characters), and a simple, practical estimator torecover the discrete spectrum from finite data. What is new. The mathematics we present is classical (Bohr, Besicovitch, Wiener,Corduneanu); the novelty here is a unified, discrete, and self-contained exposition forsequences, together with a practical spectral estimator and worked arithmetic examplessuitable for applications. We make explicit the BAP hypothesis wherever it is needed,correcting common pitfalls.