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Theorem of summability methods
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:
An example is
Cesaro summation, a matrix summability method with
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.