[Paper Review] Periodic functions with variable period

M. V Pryjmak|arXiv (Cornell University)|Jun 8, 2010

Meromorphic and Entire Functions3 citations

TL;DR

This paper introduces a mathematical framework for periodic functions with variable periods, generalizing classical trigonometric systems. It defines such functions, proves orthogonality of the resulting trigonometric system, and proposes a generalized system with conditions for existence, extending harmonic analysis to non-uniform periodicity.

ABSTRACT

The examples of rhythmical signals with variable period are considered. The definition of periodic function with the variable period is given as a model of such signals. The examples of such functions are given and their variable periods are written in the explicit form. The system of trigonometric functions with the variable period is considered and its orthogonality is proved. The generalized system of trigonometric functions with the variable period is also suggested; some conditions of its existence are considered.

Motivation & Objective

• To model real-world rhythmical signals that exhibit non-uniform periodicity, such as biological or mechanical oscillations.
• To formalize a rigorous definition of periodic functions with variable periods, moving beyond constant-period assumptions.
• To establish a system of trigonometric functions with variable periods and prove their orthogonality.
• To propose a generalized system of trigonometric functions with variable periods and identify conditions under which it exists.
• To extend classical harmonic analysis to non-stationary periodic phenomena through a mathematically consistent framework.

Proposed method

• Defines a periodic function with variable period as one where the period function T(t) varies over time, satisfying f(t + T(t)) = f(t).
• Constructs a trigonometric system using functions of the form cos(2π∫₀ᵗ dt/T(s)) and sin(2π∫₀ᵗ dt/T(s)) to match the time-varying frequency.
• Proves orthogonality of the system over a given interval by showing the inner product of distinct basis functions vanishes.
• Introduces a generalized system by extending the basis functions to include arbitrary phase and amplitude modulation under variable period.
• Analyzes existence conditions for the generalized system, focusing on integrability and regularity of the period function T(t).
• Applies functional analytic techniques to ensure completeness and orthonormality of the system under specified constraints.

Experimental results

Research questions

• RQ1 How can periodic functions with time-varying periods be formally defined to model real-world signals?
• RQ2 What conditions ensure the orthogonality of trigonometric functions when the period is not constant?
• RQ3 Can a generalized trigonometric system with variable periods be constructed, and under what mathematical conditions does it exist?
• RQ4 How does the variable period affect the decomposition of signals into harmonic components?
• RQ5 What are the implications of variable periodicity for Fourier-type analysis in non-stationary systems?

Key findings

• The paper successfully defines periodic functions with variable periods using a time-dependent period function T(t), satisfying f(t + T(t)) = f(t).
• The trigonometric system based on ∫₀ᵗ dt/T(s) is proven to be orthogonal over a finite interval, generalizing the standard Fourier basis.
• A generalized system of trigonometric functions with variable periods is proposed, with existence conditions derived from integrability and smoothness of T(t).
• The orthogonality of the system is established through direct computation of inner products, showing vanishing cross-terms.
• The framework allows for signal decomposition into harmonic components even when the fundamental frequency varies over time.
• The results provide a foundation for harmonic analysis in non-uniformly periodic systems, applicable to biological rhythms and non-stationary signals.