Silverman–Toeplitz theorem

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.[1] An infinite matrix {displaystyle (a_{i,j})_{i,jin mathbb {N} }} with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties: {displaystyle {begin{aligned}&lim _{ito infty }a_{i,j}=0quad jin mathbb {N} &&{text{(Every column sequence converges to 0.)}}\[3pt]&lim _{ito infty }sum _{j=0}^{infty }a_{i,j}=1&&{text{(The row sums converge to 1.)}}\[3pt]&sup _{i}sum _{j=0}^{infty }vert a_{i,j}vert mend{cases}}={begin{pmatrix}1&0&0&0&0&cdots \{frac {1}{2}}&{frac {1}{2}}&0&0&0&cdots \{frac {1}{3}}&{frac {1}{3}}&{frac {1}{3}}&0&0&cdots \{frac {1}{4}}&{frac {1}{4}}&{frac {1}{4}}&{frac {1}{4}}&0&cdots \{frac {1}{5}}&{frac {1}{5}}&{frac {1}{5}}&{frac {1}{5}}&{frac {1}{5}}&cdots \vdots &vdots &vdots &vdots &vdots &ddots \end{pmatrix}},} References Citations ^ Silverman–Toeplitz theorem, by Ruder, Brian, Published 1966, Call number LD2668 .R4 1966 R915, Publisher Kansas State University, Internet Archive Further reading Toeplitz, Otto (1911) "Über allgemeine 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 Hardy, G. H. (1949), Divergent Series, Oxford: Clarendon Press, 43-48. Boos, Johann (2000). Classical and modern methods in summability. New York: Oxford University Press. ISBN 019850165X. Categories: Theorems in analysisSummability methodsSummability theory