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 that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.[1]

An infinite matrix with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:

(every column sequence converges to 0)
(the row sums converge to 1)
(the absolute row sums are bounded).


  1. ^ Silverman-Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive
  • Toeplitz, Otto (1911) "Über die lineare Mittelbildungen." Prace mat.-fiz., 22, 113–118 (the original paper in German)
  • Silverman, Louis Lazarus (1913) "On the definition of the sum of a divergent series." University of Missouri Studies, Math. Series I, 1–96