A summability (Toeplitz) matrix, the transformed sequence $y_n = \sum_k c_{n,k} x_k$, and regularity

A summability matrix, also called a Toeplitz matrix, is a function

$c : \mathbb{N} \times \mathbb{N} \to \mathbb{R}, \qquad (n,k) \mapsto c_{n,k},$

with finite row support: for every $n \in \mathbb{N}$ there is $R \in \mathbb{N}$ such that $c_{n,k} = 0$ for every $k > R$. Such an $R$ is called an admissible bound for row $n$. Rows are indexed by $n$ and columns by $k$; the $k$-th column of $c$ is the sequence $n \mapsto c_{n,k}$ (Sequences of reals: bounded, eventually, frequently, tails, subsequences).

The transform. Let $(x_k)$ be a sequence of reals. The transform of $(x_k)$ by $c$ is the sequence $(y_n)$ given by

$y_n \;:=\; \sum_{k=0}^{R} c_{n,k}\,x_k ,$

where $R$ is any admissible bound for row $n$ and the sum is the finite sum of Finite sums and finite products, by recursion. We write $\sum_k c_{n,k} x_k$ for this value.

This is well defined, and the check is the reason finite row support is part of the definition. Suppose $R \le R'$ are both admissible bounds for row $n$. Splitting the longer sum (Laws of finite sums and finite products) gives

$\sum_{k=0}^{R'} c_{n,k}x_k \;=\; \sum_{k=0}^{R} c_{n,k}x_k \;+\; \sum_{k=R+1}^{R'} c_{n,k}x_k ,$

and every term of the second sum has $k > R$, hence $c_{n,k} = 0$ and $c_{n,k}x_k = 0$; a finite sum all of whose terms are $0$ is $0$, by the scaling law of Laws of finite sums and finite products with $\lambda = 0$. So the two agree. For two arbitrary admissible bounds $R_1, R_2$, the order on $\mathbb{N}$ is total ($\le$ is a linear order on $\mathbb{N}$), so each may be compared with the larger of the two, and the three values agree. Hence $y_n$ is a single well-determined real for each $n$, and $(y_n)$ is a sequence of reals.

Two instances of the transform have their own names. The row sum of row $n$ is $\sum_k c_{n,k}$, the transform of the constant sequence $1$; the row absolute sum is $\sum_k |c_{n,k}|$, the transform of the constant sequence $1$ by the matrix $(n,k) \mapsto |c_{n,k}|$, which again has finite row support with the same admissible bounds.

Regularity. The summability matrix $c$ is regular when for every convergent sequence $(x_k)$ of reals the transform $(y_n)$ converges and

$\lim_n y_n \;=\; \lim_k x_k .$

Both limits are asserted to exist there: the right-hand one by hypothesis on $(x_k)$, the left-hand one as part of the condition. Limits of real sequences are unique (A sequence has at most one limit, Limits and Cauchy sequences of reals), so the condition is unambiguous.

• Finite row support is not a technical convenience, it is what makes the transform mean anything at all at this point of the library. The classical definition allows every row to be an infinite series $\sum_{k=0}^{\infty} c_{n,k}x_k$, and asks that each such series converge. Series are not defined anywhere in this library yet; they arrive on the next page of this track. Every sum above is therefore a finite sum in the sense of Finite sums and finite products, by recursion, and no convergence question arises inside a row.

• What is lost, and what is not. The restriction excludes matrices such as the Abel and Borel means, whose rows are genuine series. It does not exclude anything needed here: the Cesaro matrix has $c_{n,k} = 1/(n+1)$ for $k \le n$ and $0$ beyond (The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem), so row $n$ has admissible bound $n$; and the counterexample of A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one ↗ has two nonzero entries per row. A summability matrix with only finitely many nonzero entries per row is regular iff each column tends to $0$, the row sums tend to $1$, and the row absolute sums are uniformly bounded characterises regularity within this class.

• Regularity says the transform is a genuine generalisation of the limit. It is exactly the condition that $c$ never changes the value of a limit that already exists. It says nothing at all about sequences that do not converge, and the interest of such matrices is precisely that a regular one may still assign a value to a divergent sequence: the Cesaro matrix does so for the alternating sequence (The Cesaro means of $(-1)^k$ converge to $0$ although the sequence diverges ↗).

• A matrix that fails regularity can fail it in two different ways, by changing a limit or by destroying convergence outright. A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one ↗ does the second, which is the stronger failure.