Theorem of summability methods
In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transformation that preserves the limits of convergent sequences. The linear sequence transformation can be applied to the divergent sequences of partial sums of divergent series to give those series generalized sums.
An infinite matrix formula_1 with complex-valued entries defines a regular matrix summability method if and only if it satisfies all of the following properties:
 formula_2
An example is Cesàro summation, a matrix summability method with
formula_3
Formal statement.
Let the aforementioned inifinite matrix formula_1 of complex elements satisfy the following conditions:
and formula_8 be a sequence of complex numbers that converges to formula_9. We denote formula_10 as the weighted sum sequence: formula_11.
Then the following results hold:
Proof.
Proving 1..
For the fixed formula_17 the complex sequences formula_8, formula_10 and formula_20 approach zero if and only if the real-values sequences formula_21, formula_22 and formula_23 approach zero respectively. We also introduce formula_24.
Since formula_25, for prematurely chosen formula_26 there exists formula_27, so for every formula_28 we have formula_29. Next, for some formula_30 it's true, that formula_31 for every formula_32 and formula_33. Therefore, for every formula_32
formula_35
which means, that both sequences formula_22 and formula_10 converge zero.
Proving 2..
formula_38. Applying the already proven statement yields formula_39. Finally,
formula_40, which completes the proof.
References.
Citations.
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