What question did this study set out to answer?

The aim is to introduce and formalize the Theseus Expansion for constructing smooth functions.

April 17, 20260 citationsOpen Access

Theseus Expansion

Key Points

The aim is to introduce and formalize the Theseus Expansion for constructing smooth functions.
Introduce a functional framework for constructing functions sequentially.
Use local generating functions for patchwise assembly.
Prove smoothness theorems to ensure continuity and differentiability.
Establishes the flexibility of local functional design compared to classical methods.
Demonstrates preservation of smoothness at subinterval boundaries.
Explores applications in various fields like numerical analysis and signal processing.

Abstract

The Theseus Expansion is introduced as a novel functional framework thatallows the construction of globally smooth functions via sequential, patchwise assembly of local generating functions. Unlike classical expansions such as Taylor orFourier series, which rely on global smoothness or analyticity, the Theseus Expansion provides flexibility in local functional design while preserving continuity anddifferentiability at subinterval boundaries. This paper formalizes the theoreticalfoundations, proves smoothness theorems, and explores potential applications innumerical analysis, signal processing, and adaptive modeling.

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