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On some properties of summability methods with variable order

Georgian Mathematical Journal, 2017
Methods of summability with variable order are considered. A connection of series convergence with series summability is established. A non-strengthening in a certain sense of some results is proved.
Shakro Tetunashvili
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Consistent Summability Methods

Journal of the London Mathematical Society, 1957
Es werden zwei miteinander verträgliche (consistent), positive und permanente Matrixverfahren \(A\) und \(B\) konstruiert, zu denen es kein beide enthaltendes positives und permanentes Matrixverfahren \(C\) gibt. Die Konstruktion beruht auf folgendem Gedanken: Wenn \(A\) die Folge \(\{\sigma_n^1\} = \{1, 0, 1, 0, \ldots\}\) zu \(\frac14\) und \(B\) die
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On Scales of Summability Methods

Mathematische Nachrichten, 1995
AbstractIn this paper we consider generalized Nörlund methods (Nαp), α > ‐1, power series methods (Jp) and the iteration product of two such methods. A particular case is that of the Cesaro means (Cα) with corresponding power series method (A), i.e., Abel's method.
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Potent Conservative Summability Methods

Bulletin of the London Mathematical Society, 1994
The theorems of this paper are given in a very short form. Before stating them, we need to have some notation and definitions. Notation: \(\chi\) -- the set of all sequences of 0s and 1s; \(K\) -- the set of all conservative matrices; \(T\) -- the set of all thin sequences; \(F\) -- the set of all almost convergent sequences; \((M)\) -- the set of ...
Kuttner, Brian, Parameswaran, Mangalam R.   +1 more
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ON SOME ABSOLUTE SUMMABILITY METHODS

Analysis, 1987
Let \(\Sigma a_ n\) be a given infinite series with sequence of partial sums \(\{s_ n\}\). The series \(\Sigma a_ n\) is said to be summable \(| \bar N,P_ n|_ k,k\geq 1\), if \(\sum^{\infty}_{n=1}(P_ n/p_ n)^{k-1}| t_ n-t_{n-1}|^ k0\), \(P_ n=p_ 0+p_ 1+...+p_ n\to \infty\) and \(t_ n=P_ n^{- 1}\sum^{n}_{\nu =0}p_{\nu}S_{\nu}.\) In the special case when
Bor, H., Thorpe, B.
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Direct Theorems on Methods of Summability

Canadian Journal of Mathematics, 1949
1.1. A regular Toeplitz method of summability is given by a transformation m = 0, 1, 2 , … of the sequence sn into the sequence σm. According to the definition of regularity, every such method sums a convergent sequence sn to the value lim sn. The question naturally arises, whether there are more extensive classes of sequences summable by all regular
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On Riesz and Cesàro methods of summability

Transactions of the American Mathematical Society, 1933
* Presented to the Society, December 30, 1931; received by the editors August 24, 1932. t National Research Fellow. t Comptes Rendus, vol. 149 (1909), pp. 18-22. In this note Riesz considered only real positive orders r. ? Comptes Rendus, vol. 152 (1911), pp. 1651-1654. Here again Riesz considered only the case r>O.
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On Borel‐type methods of summability

Mathematika, 1958
Suppose throughout that l, a n ( n = 0, 1, …) are arbitrary complex numbers, that α is a fixed positive number and that x is a variable in the interval [0,µ ...
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On a summability method

1979
In previous studies we applied Lanzcos' τ-method to get polynomial and rational approximations to series of hypergeometric type. It was shown that the approximations could be viewed as a weighted sum of the partial sums of the given series. This we call a summability method.
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Scales of Logarithmic Methods of Summability

Canadian Mathematical Bulletin, 1969
We suppose throughout that p is a non-negative integer, and use the following notations:where log0x = x for x ≥ e0 = 1, and logn+1x = log(lognx) for x ≥ en+1 = een (n = 0, 1, 2,…);
Borwein, D., Phillips, R.
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