Two Computable Invariants for Fourier Approximation of Piecewise-Smooth Signals

Key Points

• The aim is to establish two diagnostics for assessing Fourier approximations of piecewise-smooth periodic signals with discontinuities.
• Introduces two invariants: energy concentration invariant, radius budget invariant.
• Presents theoretical statements and asymptotic mechanisms for analysis.
• Uses numerical verification for specific signal types including square-wave, sawtooth, and triangle controls.
• Demonstrates the effectiveness of the proposed invariants in assessing Fourier approximations.
• Provides falsifiable criteria to verify the diagnostics' accuracy.
• Shows numerical examples that validate the theoretical claims.