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Frobenius' theorem: Cesaro summability of a real series implies Abel summability to the same value

Statement

If a real series is Cesaro summable to ss, then it is Abel summable to ss.

Facts & Assumptions

Given: Cesaro means σn→s\sigma_n\to s for the partial sums of ∑an\sum a_n.

[L1]

If (σn)(\sigma_n) is bounded, the Abel series converges for 0<x<10<x<1 and its transform is A(x)=(1−x)2∑n≥0ι(n+1)σnxnA(x)=(1-x)^2\sum_{n\ge0}\iota(n+1)\sigma_nx^n (For 0<x<10<x<1, the Abel transform of a series is (1−x)2∑n≥0(n+1)σnxn(1-x)^2\sum_{n\ge0}(n+1)\sigma_nx^n, where σn\sigma_n are the Cesaro means of its partial sums).

[L2]

The nonnegative weights wn(x):=(1−x)2ι(n+1)xnw_n(x):=(1-x)^2\iota(n+1)x^n sum to 11 for 0<x<10<x<1. Indeed, apply the transform in [L1] to the series with coefficients 1,0,0,…1,0,0,\ldots, whose partial sums and Cesaro means are all 11 (For 0<x<10<x<1, the Abel transform of a series is (1−x)2∑n≥0(n+1)σnxn(1-x)^2\sum_{n\ge0}(n+1)\sigma_nx^n, where σn\sigma_n are the Cesaro means of its partial sums).

[L3]

Abel summability to ss means that the Abel series converges on 0≤x<10\le x<1 and tends to ss as x↑1x\uparrow1 (Abel summability by lim⁡x↑1∑anxn\lim_{x\uparrow1}\sum a_nx^n and Cesaro summability by the Cesaro means of the partial sums).

Proof

technique · direct
1.1

Since (σn)(\sigma_n) converges, it is bounded, so [L1] applies and A(x)−s=∑n≥0wn(x)(σn−s)A(x)-s=\sum_{n\ge0}w_n(x)(\sigma_n-s).

givenL1L2
2.1

Given ε>0\varepsilon>0, choose NN with ∣σn−s∣<ε|\sigma_n-s|<\varepsilon for n≥Nn\ge N. The corresponding tail is at most ε∑n≥Nwn(x)≤ε\varepsilon\sum_{n\ge N}w_n(x)\le\varepsilon.

step 1.1L2choose
3.1

For each fixed nn, wn(x)→0w_n(x)\to0 as x↑1x\uparrow1, so the finite head tends to 00. Together with step 2.1 this gives A(x)→sA(x)\to s, which is Abel summability by definition.

step 1.1step 2.1L2L3∎

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