The Atomic Decomposition Hypothesis of Basic Functions and Mathematical Structures: A Unified Perspective from Prime Numbers to Functional Analysis

Author: Zhang Suhang

Affiliation: Independent Researcher, Luoyang

Abstract

Starting from an intuitive and profound mathematical insight—that complex functions can be constructed via linear combinations of a set of "basic functions," analogous to the unique composite representation of integers by prime numbers in the Fundamental Theorem of Arithmetic—this paper proposes a unifying hypothesis for mathematical structures. By systematically examining Taylor series, Fourier series, and orthogonal basis theory in functional analysis, it argues for the universality and necessity of an underlying "atomic basis" in function spaces, thereby fully revealing the structural isomorphism from discrete number theory to continuous analysis. Without relying on external axiomatic systems, and grounded solely in established results of classical analysis and functional analysis, this paper independently proposes and elaborates the hypothesis of atomic decomposition of functions. Its aim is to uncover the deep unity among different branches of mathematics, providing a concise, self-consistent, and heuristic unified cognitive framework for the theory of function representation.

Keywords: Basic functions; Prime number analogy; Atomic decomposition; Functional analysis; Hilbert spaces; Function representation; Structural unity

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1 Introduction

Different branches of mathematics often conceal highly consistent underlying structures. Among the most representative is the parallel correspondence between discrete number theory and continuous analysis:

· The Fundamental Theorem of Arithmetic: Every integer greater than 1 can be uniquely decomposed into a finite product of prime numbers. Primes are the "indivisible atoms" of the integer system.

· Classical methods in analysis: Any sufficiently regular function can be expanded as a power series, a trigonometric series, or as a linear combination of other canonical basis functions. These basis functions are the "fundamental building blocks" of function spaces.

Based on this clear structural analogy, this paper independently proposes the Atomic Decomposition Hypothesis for Functions: There exists a well-defined class of "basic functions" that constitute an atomic basis for the corresponding function space, enabling any function satisfying certain mild regularity conditions to be uniquely and exactly represented as a linear combination of these basic functions.

Grounding itself entirely in established classical mathematics and functional analysis, independent of external axioms and without additional presuppositions, this hypothesis reveals the deep isomorphism between continuous function systems and the integer number system in terms of the "atom–composition" logic, by inducing and refining the structural laws of mathematics itself. It aims to awaken an intuitive understanding of the overall unity of mathematics.

2 Historical Context: Embryonic Atomic Representations by Taylor and Fourier

2.1 Taylor Series: Power Functions as a Naive Atomic Basis

If a single-variable function f(x) is infinitely differentiable in a neighborhood of point a , it can be expanded as a Taylor series:

f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^n.

Here, the family of basic functions is \{1, (x-a), (x-a)^2, (x-a)^3, \dots\} , each term being an indivisible power-function unit, with coefficients uniquely determined by the derivative information of the original function. This is the most straightforward and naive realization, within continuous analysis, of the idea that complex objects are constructed linearly from simple atoms.

2.2 Fourier Series: Trigonometric Functions as a Canonical Orthonormal Atomic Basis

For a square-integrable function with period 2\pi , the Fourier expansion yields:

f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} \big(a_n \cos nx + b_n \sin nx\big).

In this expansion, the basis functions \{\cos nx, \sin nx \mid n \in \mathbb{N}\} form a standard orthonormal basis for the Hilbert space L^2[0, 2\pi] . Compared to Taylor series, the Fourier series extended the notion of "basic functions" from power functions to an abstract family of functions equipped with orthogonality and normalization. This marked a transition in the atomic representation of functions from a "formal expansion" to a rigorous theory grounded in spatial structure.

3 The Prime Number Analogy: A Unified Atomism of Mathematical Structures

The integer system and function spaces exhibit a perfect structural isomorphism in their underlying construction logic:

Integer Number System (Discrete) Function Spaces (Continuous)

Basic atoms: prime numbers p_1, p_2, p_3, \dots , indivisible and generating all integers Basic atoms: basic functions \phi_1, \phi_2, \phi_3, \dots , irreducible and completely generating the space

Composition rule: finite product Composition rule: infinite linear combination (in the sense of convergent series)

Core theorem: The Fundamental Theorem of Arithmetic (existence and uniqueness of decomposition) Core hypothesis: The Atomic Decomposition Hypothesis for Functions (uniqueness of representation for sufficiently regular functions)

The difference lies only in the form of the operation: product-type discrete decomposition for integers, versus linear combination-type continuous expansion for functions. This distinction precisely mirrors the paradigmatic divide between discrete and continuous mathematics. Yet the core logic—"atoms constructing complex wholes"—remains entirely consistent. This consistency provides the most solid intuitive support for the hypothesis advanced in this paper.

4 Rigorous Theoretical Support from Functional Analysis

The axiomatic framework of 20th-century functional analysis provides a complete and rigorous mathematical foundation for the Atomic Decomposition Hypothesis for Functions. This elevates the hypothesis from an intuitive analogy to a theoretical proposition amenable to rigorous validation:

· The theory of standard orthonormal bases in Hilbert spaces: Families such as Legendre polynomials, Hermite polynomials, Laguerre polynomials, and wavelets are essentially "basic function atomic families" for their respective function spaces, satisfying completeness, orthogonality, and irreducibility.

· The Spectral Theorem: The eigenfunctions of a self-adjoint linear operator can form a canonical basis for the space—i.e., a "natural atomic basis" tailored to the operator's action.

· Basis theory in Banach spaces: Schauder bases in separable Banach spaces further extend the atomic representation from Hilbert spaces to more general normed linear spaces, broadening the scope of the hypothesis.

The above achievements demonstrate that the Atomic Decomposition Hypothesis for Functions proposed in this paper is not a mere formal analogy, but a systematic induction of deep