Silverman–Toeplitz theorem

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.

An infinite matrix ( a i , j ) i , j ∈ N with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties:

lim i → ∞ a i , j = 0 j ∈ N (every column sequence converges to 0) lim i → ∞ ∑ j = 0 ∞ a i , j = 1 (the row sums converge to 1) sup i ∑ j = 0 ∞ | a i , j | < ∞ (the absolute row sums are bounded).