Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

Summation Chains of Sequences Part 3: Sequence Chains from Linear Functions

Abstract

(Editor’s note:) This paper represents the third installment of a masters thesis by Jonathan Johnson. The first two can be found here and here. This paper continues the development of the theory of summation chains of sequences. Since summation chains are doubly infinite, it’s important to know how little information we actually need to define a chain. The linearity of the function rules that generates a summation chain helps to answer this question. The notion of uniquely completable is defined from the set of positions, and several important theorems are developed to determine when a set of positions is uniquely completeable.

Introduction

(Editor’s note:) In Part 2, Johnson notes that summation chains can be generated by the function rule $T_{\Sigma}: M^{\infty} \to M^{\infty}$,
$$T_{\Sigma}(x_{1}, x_{2}, x_{3},\ldots) = (x_{1}, x_{1} + x_{2}, x_{1} + x_{2} + x_{3},\ldots).$$ where $M^{\infty}$ is a $\mathbb{Z}$-Module. This is a formal way to define the operator that generates the partial sums of a given sequence. Johnson proves in this installment that this function rule is a linear operator. The linearity of these function operators assists in determining how little information we need to know about the chain to be able to uniquely define it.

Johnson introduces the notion of the set of positions, which becomes the smallest amount of information that can define a summation chain under certain conditions.

Chains Generated by Linear Functions

Since vector-valued chains are vector-valued functions, they inherit the scalar multiplication and addition operations defined for functions on vector spaces. The linearity of the summation chain function rule leads to many interesting and useful results. The first of these is the closure of the set of summation chains under linear operations. This result holds for any function rule that is a linear operator.

Lemma.

Let M be a an 1-dimensional vector space with scalar field F. The summation function rule $T_{\Sigma}$ defined in Remark 3.1 of [6] is a linear operator on $M^{\infty}$.

Proposition.

Let $T:V\to V$ be a function rule where V is a vector space with scalar field F.
$\mathcal{C}_T$ is closed under scalar multiplication and addition if and only if T is a linear operator.

Corollary 1.

Let $T:V\to V$ be a bijective linear operator on vector space V with scalar field F, then $\mathcal{C}_T$ is a subspace of the space of all functions from $\mathbb{Z}$ to V, $V^{\mathbb{Z}}$, $\Phi_{\Sigma[M]}$ is an isomorphism, and $\mathcal{C}_T\cong V$.

Corollary 2.

Let M be a an 1-dimensional vector space with scalar field F.  $\mathcal{C}_{\Sigma[M]}$ is a vector space, and $\Phi_{\Sigma[M]}$ is an isomorphism from $M^{\infty}$ to $\mathcal{C}_{\Sigma[M]}$.