Silverman–Toeplitz theorem
Encyclopedia
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
.
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).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz
Otto Toeplitz
Otto Toeplitz was a German Jewish mathematician working in functional analysis.- Life and work :...
, is a result in summability theory characterizing matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
.
Silverman–Toeplitz theorem
An infinite matrix with complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
-valued entries defines a regular summability method if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
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).