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In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in series summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a linear sequence transformation that preserves the limits of convergent sequences . [ 1 ]
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Symmetric Toeplitz matrices are both centrosymmetric and bisymmetric. Toeplitz matrices are also closely connected with Fourier series, because the multiplication operator by a trigonometric polynomial, compressed to a finite-dimensional space, can be represented by such a matrix. Similarly, one can represent linear convolution as ...
Toeplitz algebra, the C*-algebra generated by the unilateral shift on the Hilbert space; Toeplitz Hash Algorithm, used in many network interface controllers; Hellinger–Toeplitz theorem, an everywhere defined symmetric operator on a Hilbert space is bounded; Silverman–Toeplitz theorem, characterizing matrix summability methods which are regular
The Silverman–Toeplitz theorem characterizes matrix summation methods, which are methods for summing a divergent series by applying an infinite matrix to the vector of coefficients. The most general methods for summing a divergent series are non-constructive and concern Banach limits .
In mathematical analysis, the Szegő limit theorems describe the asymptotic behaviour of the determinants of large Toeplitz matrices. [ 1 ] [ 2 ] [ 3 ] They were first proved by Gábor Szegő . Notation
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Let () = be a sequence, and let = + + = = be its k th partial sum.. The sequence (a n) is called Cesàro summable, with Cesàro sum A ∈ , if, as n tends to infinity, the arithmetic mean of its first n partial sums s 1, s 2, ..., s n tends to A: