Silverman–Toeplitz theorem
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In mathematics the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods which are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.
[edit] Silverman–Toeplitz theorem
An infinite matrix with complex valued entries defines a regular summability method if and only if it satisfies all of the following properties
- (any column sequences converge to 0)
- (the sum of the rows converges to 1)
- (the row sums are bounded)
[edit] References
- Toeplitz, O. (1911) "Über die lineare Mittelbildungen" Prace mat.-fiz., 22, 113-118 (the original paper in German)