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The Cesaro matrix satisfies the Silverman-Toeplitz conditions, giving a second proof of the Cesaro mean theorem

Statement

Define $c_{n, k} : = (n + 1)^{- 1}$ for $k \leq n$ and $c_{n, k} : = 0$ for $k > n$ . Then:

$c$ is a summability matrix (A summability (Toeplitz) matrix, the transformed sequence $y_{n} = \sum_{k} c_{n, k} x_{k}$ , and regularity), with $n$ an admissible bound for row $n$ ;
the transform of a sequence $(x_{k})$ by $c$ is exactly its sequence of Cesaro means (The Cesaro means $σ_{n} = (x_{0} + \dots + x_{n}) / (n + 1)$ and $(C, 1)$ -summability), $y_{n} = σ_{n}$ ;
$c$ satisfies the three conditions of 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, so $c$ is regular.

Consequently every convergent sequence has $σ_{n} \to \lim_{k} x_{k}$ , which is a second proof of If $x_{k} \to L$ then $σ_{n} \to L$ : convergence implies $(C, 1)$ -summability to the same value, obtained from the general characterisation rather than from a direct estimate.

Facts & Assumptions

Given: The matrix $c$ with $c_{n, k} = (n + 1)^{- 1}$ for $k \leq n$ and $c_{n, k} = 0$ for $k > n$ .

[L1]

Summability matrices: finite row support, the transform, the row sum, the row absolute sum and regularity (A summability (Toeplitz) matrix, the transformed sequence $y_{n} = \sum_{k} c_{n, k} x_{k}$ , and regularity, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[L2]

The Cesaro means $σ_{n} = (n + 1)^{- 1} \sum_{k = 0}^{n} x_{k}$ (The Cesaro means $σ_{n} = (x_{0} + \dots + x_{n}) / (n + 1)$ and $(C, 1)$ -summability).

[L3]

Silverman-Toeplitz: a summability matrix is regular exactly when every column tends to $0$ , the row sums tend to $1$ , and the row absolute sums are uniformly bounded (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).

[L4]

Finite sums and their laws, in particular $\sum_{k < d} λ = d λ$ (Finite sums and finite products, by recursion, Laws of finite sums and finite products).

[L5]

Convergence, and the fact that a constant sequence converges to its value (Limits and Cauchy sequences of reals, Sequences of reals: bounded, eventually, frequently, tails, subsequences).

[L6]

Reciprocal Archimedean property: for every real $ε > 0$ there is a natural $m \geq 1$ with $1 / m < ε$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L7]

Order arithmetic: $(n + 1) \cdot 1_{R} > 0$ for every $n \in N$ (Canonical naturals are positive and strictly increasing); a positive element is invertible with positive inverse, and reciprocation reverses the order (Inverses of positives are positive, and reciprocation reverses order); $∣ u ∣ = u$ for $u \geq 0$ (Basic properties of the absolute value); the order is total and transitive (Complete ordered field (least-upper-bound property), Ordered field).

Proof

technique · direct

1.1

Row $n$ of $c$ vanishes at every $k > n$ , so $n$ is an admissible bound for row $n$ and $c$ is a summability matrix; its transform is $y_{n} = \sum_{k = 0}^{n} (n + 1)^{- 1} x_{k} = (n + 1)^{- 1} \sum_{k = 0}^{n} x_{k} = σ_{n}$ .

L1L2L4

1.2

For every $n$ the canonical natural $(n + 1) \cdot 1_{R}$ is positive, hence invertible with $(n + 1)^{- 1} > 0$ ; so $∣ c_{n, k} ∣ = c_{n, k}$ for all $n, k$ .

2.1

Columns are null. Fix $k$ and let $ε > 0$ ; choose $m \geq 1$ with $1 / m < ε$ . For $n \geq m$ one has $n + 1 > m$ , so $∣ c_{n, k} - 0 ∣ \leq (n + 1)^{- 1} < 1 / m < ε$ , the case $n < k$ giving $c_{n, k} = 0$ outright. Hence $\lim_{n} c_{n, k} = 0$ .

step 1.2L5L6L7

2.2

Row sums tend to $1$ . For every $n$ , $\sum_{k} c_{n, k} = \sum_{k = 0}^{n} (n + 1)^{- 1} = (n + 1) (n + 1)^{- 1} = 1$ , a constant sequence, which converges to $1$ .

step 1.1step 1.2L1L4L5

2.3

Row absolute sums are uniformly bounded. For every $n$ , $\sum_{k} ∣ c_{n, k} ∣ = \sum_{k} c_{n, k} = 1 \leq 1$ .

step 1.1step 1.2L1L4

3.1

All three conditions hold, so $c$ is regular.

step 2.1step 2.2step 2.3L3

4.1

Therefore, for every convergent sequence $(x_{k})$ , the transform $(σ_{n})$ converges with $\lim_{n} σ_{n} = \lim_{k} x_{k}$ .

step 1.1step 3.1L1∎

Remarks

Which condition is doing what. For the Cesaro matrix the row sums and the row absolute sums are not merely convergent and bounded, they are constantly $1$ ; the whole content is that the columns are null, that is, that each single term contributes a weight $1 / (n + 1)$ which fades away. That is the precise sense in which averaging forgets any finite head, and it is why FALSE: if the Cesaro means of a sequence converge then the sequence converges is false: forgetting the head is not the same as recovering the sequence.
Two proofs, two costs. If $x_{k} \to L$ then $σ_{n} \to L$ : convergence implies $(C, 1)$ -summability to the same value is proved directly by a head-and-tail estimate, with no machinery at all; 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 proves the same estimate once for every weighting and then reads the Cesaro case off three trivial verifications. Both are kept, because the direct proof is what a reader should see first and the general one is what generalises.
A weighting with unbounded row absolute sums need not be regular, and the Cesaro matrix is as far from that as possible, its rows being nonnegative and summing to $1$ . See A summability matrix failing exactly one Silverman-Toeplitz condition and transforming a convergent sequence to a divergent one ↗ for the contrast.

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Sources

Toeplitz matrix (Encyclopedia of Mathematics) (standard reference, not scraped)
Summation methods (Encyclopedia of Mathematics) (standard reference, not scraped)
Silverman-Toeplitz theorem (Wikipedia) (standard reference, not scraped)
Cesàro summation (Wikipedia) (standard reference, not scraped)