A Deterministic Hierarchical Smooth–Oscillatory Representation of Discrete Sequences

This paper studies a deterministic exact decomposition of a discrete sequence into asmooth component and an oscillatory component. The emphasis is not on the novelty ofthe elementary operations involved—finite differences, discrete accumulation, and fixedreconstruction rules are classical—but on the viewpoint that such operations can beorganized into an exact hierarchical representation of the sequence. Once a deterministicoperator is fixed, the original sequence is recovered exactly as the sum of a first smoothlayer and a first oscillatory residual; the same construction may then be iterated on theresidual, yielding an exact representation by successive smooth layers and a final residualof chosen order. The framework is interpreted both as a discrete representation principleand as a natural analysis–synthesis scheme, with possible conceptual connections totrend–residual decomposition, empirical mode decomposition, multiscale signal analysis,and arithmetic experimentation. Illustrative examples are discussed for polynomialsequences, factorial growth, Fibonacci numbers, prime numbers, the prime countingfunction, and sampled sinusoidal signals.