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On Some Triangular Summability Methods
American Journal of Mathematics, 1947It is to be noted that Bn (x) is defined by taking the first n + 1 terms of the series defining the function *J(x) ; the summability method is then constructed with the sequence {x.n}. As set forth in the aforementioned paper of Szasz,2 the regularity of either method (1. 1) or (1. 2) does not imply the regularity of the other method. On the other hand,
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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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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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On Borel‐type methods of summability
Mathematika, 1958Suppose 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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Scales of Logarithmic Methods of Summability
Canadian Mathematical Bulletin, 1969We 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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On the Summability of Series by a Method of Valiron
Proceedings of the Edinburgh Mathematical Society, 1936The method of summability with which I shall be concerned here is denoted by (V, α ) and is defined as follows:—The series Σαn is said to be summable (V, α ) to the sum s ifThis is a particular case of a method due to Valiron in which μ–2α is replaced by a function of μ.
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Summability Methods on Matrix Spaces
Canadian Journal of Mathematics, 1961The matrix spaces under consideration are the four main types of irreducible bounded symmetric domains given by Cartan (5). Let z = (zjk) be a matrix of complex numbers, z' its transpose, z* its conjugate transpose and I = I(n) the identity matrix of order n.
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ERGODIC THEOREMS AND SUMMABILITY METHODS
The Quarterly Journal of Mathematics, 1987Given: a regular summation method \((a_{n,m})_{n,m}\) such that \(\sum^{\infty}_{k=m}| a_{n,m+1}-a_{n,m}| \to 0\) uniformly in n and a sequence \((T_ n)_ n\) of bounded operators, chosen independently on a Banach space X. The author investigates conditions under which \(\lim_{n\to \infty}\sum^{\infty}_{m=1}a_{n,m}T_ m,...,T_ 1(x)\) (x\(\in X)\) exists ...
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Summability Methods in Perturbation Theory
Journal of Mathematical Physics, 1970The Mittag-Leffler summability method is applied to operator-valued analytic functions and a corresponding procedure for perturbation theory is derived, which has a bigger region of convergence. This region is explicitly described.
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