A General Tauberian Condition that Implies Euler Summability

1994 ◽  
Vol 37 (3) ◽  
pp. 393-398
Author(s):  
Mangalam R. Parameswaran
Keyword(s):  
Euler Method ◽  
Summability Method ◽  
Real Sequence ◽  
Tauberian Condition ◽  
Complex Sequence ◽  
The Matrix

AbstractLet V be any summability method (whether linear or conservative or not), 0 < p < 1 and s a real or complex sequence. Let Ep denote the matrix of the Euler method. A theorem is proved, giving a condition under which the V-summability of Eps will imply the Ep-summability of s. This extends, in generalized form, an earlier result of N. H. Bingham who considered the case where s is a real sequence and V = B (Borel's method). It is also proved that even for real sequences, the condition given in the theorem cannot be replaced by the condition used by Bingham.

scholarly journals A Tauberian Theorem Concerning Borel-Type and Cesàro Methods of Summability

1988 ◽  
Vol 40 (1) ◽  
pp. 228-247 ◽  
Author(s):  
David Borwein ◽  
Tom Markovich
Keyword(s):  
Tauberian Theorem ◽  
Summability Method ◽  
Negative Integer ◽  
Real Sequence ◽  
Real Numbers ◽  
Tauberian Condition ◽  
Cesaro Summability ◽  
Exponential Method ◽  
Image Position ◽  
Borel Type

Suppose throughout that r ≧ 0, α > 0, αq + β > 0 where q is a non-negative integer. Let {sn} be a sequence of real numbers,The Borel-type summability method (B, α, β) is defined as follows:The method (B, α, β) is regular [5]; and (B, 1, 1) is the standard Borel exponential method B. For a real sequence {sn} we consider the slowly decreasing-type Tauberian conditionWe shall also be concerned with the Cesàro summability method Cp(p > —1), the Valiron method Vα, and the Meyer-König method Sa (0 < a < 1) defined as follows:

Rank, trace, eigenvalues and norms of a structured matrix

2019 ◽  
Vol 12 (06) ◽  
pp. 2040015
Author(s):  
Ahmet İpek
Keyword(s):  
Real Sequence ◽  
Simple Property ◽  
Structured Matrix ◽  
The Matrix ◽  
Matrix Formula ◽  
Appl Math ◽  
Real Matrices

The paper deals with rank, trace, eigenvalues and norms of the matrix [Formula: see text], where [Formula: see text] are ith components of any real sequence [Formula: see text]. A result in this paper is that the Euclidean and spectral norms of the matrix [Formula: see text] is [Formula: see text]. This is a generalization of the main result by Solak [Appl. Math. Comput. 232 (2014) 919–921], with the proof based on a simple property of norms of real matrices.

A counter-example in summability theory

1978 ◽  
Vol 83 (3) ◽  
pp. 353-355
Author(s):  
B. Kuttner
Keyword(s):  
Summability Method ◽  
Complex Numbers ◽  
Tauberian Condition ◽  
Counter Example ◽  
Image Position ◽  
Regular Summability ◽  
Natural Way

Let Σ denote the set of all seriesof complex numbers. By a ‘summability method’, say A, we mean a function from some subset (the set of ‘A -summable series’) of Σ into the set of complex numbers. We will use the language generally associated with this definition, and will take for granted the case in which A is (C, 1). A summability method A will be called linear if, whenever a, b are A -summable, then so is λa + μb (where λ, μ are any complex constants) and if the. A -sums of a, b, λa + μb are then related in the natural way. We call A regular if, whenever a converges to σ, it is A -summable to σ. If A is a regular summability method, then any condition P on the series (1) will be called a Tauberian condition for A if any A -summable series which satisfies P is convergent.

Absolute summability functions for Hausdorff methods

1982 ◽  
Vol 91 (1) ◽  
pp. 51-56
Author(s):  
B. Kuttner
Keyword(s):  
Absolute Summability ◽  
Bounded Sequence ◽  
Summability Method ◽  
Negative Integer ◽  
Hausdorff Method ◽  
The Matrix ◽  
Sequence Transformation ◽  
Image Position

1. Following Lorentz, we suppose throughout that Ω(n) is a non-negati ve non-decreasing function of the non-negative integer n such that Ω(n)→ ∞ as n → ∞. Consider the summability method given by the sequence-to-sequence transformation corresponding to the matrix A = (ank). We say that Ω(n) is a summability function for A (or absolute summability function for A) if the following holds: Any bounded sequence {sn} such that the number of values of ν with ν ≤ n, sν ≠ 0 does not exceed Ω(n) is summable A (or is absolutely summable A, respectively). These definitions are due to Lorentz (4), (6). We shall be concerned with the case in which A is a regular Hausdorff method, say A = H = (H, μn). Then H is given by the matrix (hnk) withwithX(0) = X(0 + ) = 0, X(1) = 1;(see e.g.(1), chapter XI). We shall suppose throughout that these conditions are satisfied. It is known that H is then necessarily also absolutely regular.

A Tauberian Theorem for the General Euler-Borel Summability Method

1992 ◽  
Vol 44 (5) ◽  
pp. 1100-1120 ◽  
Author(s):  
Laying Tam
Keyword(s):  
Positive Integer ◽  
Unit Disk ◽  
Tauberian Theorem ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Borel Summability ◽  
Closed Unit Disk ◽  
Tauberian Condition ◽  
Image Position ◽  
General Conditions

AbstractOur main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.

scholarly journals Tauberian conditions with controlled oscillatory behavior for statistical convergence

Filomat ◽  
2018 ◽  
Vol 32 (11) ◽  
pp. 3853-3865
Author(s):  
Sefa Sezer ◽  
Rahmet Savaş ◽  
İbrahim Çanak
Keyword(s):  
Finite Number ◽  
Statistical Convergence ◽  
Oscillatory Behavior ◽  
Summability Method ◽  
Tauberian Theorems ◽  
Real Sequence ◽  
Tauberian Conditions

We present new Tauberian conditions in terms of the general logarithmic control modulo of the oscillatory behavior of a real sequence (sn) to obtain lim n?? sn = ? from st - lim n?? sn = ?, where ? is a finite number. We also introduce the statistical (l,m) summability method and extend some Tauberian theorems to this method. The main results improve some well-known Tauberian theorems obtained for the statistical convergence.

scholarly journals Vector-valued sequence spaces generated by infinite matrices

2001 ◽  
Vol 26 (9) ◽  
pp. 547-560
Author(s):  
Nandita Rath
Keyword(s):  
Sequence Spaces ◽  
Infinite Matrix ◽  
Real Sequence ◽  
Normed Spaces ◽  
Infinite Matrices ◽  
Positive Real ◽  
The Matrix ◽  
Almost All ◽  
Vector Valued

LetA=(ank)be an infinite matrix with allank≥0andPa bounded, positive real sequence. For normed spacesEandEkthe matrixAgenerates paranormed sequence spaces such as[A,P]∞((Ek)),[A,P]0((Ek)), and[A,P](E)which generalize almost all the existing sequence spaces, such asl∞,c0,c,lp,wp, and several others. In this paper, conditions under which these three paranormed spaces are separable, complete, andr-convex, are established.

scholarly journals Accurate Numerical Method for Pricing Two-Asset American Put Options

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Xianbin Wu
Keyword(s):  
Time Derivative ◽  
Euler Method ◽  
Step Size ◽  
Central Difference ◽  
Put Options ◽  
The Maximum Principle ◽  
American Put Options ◽  
The Matrix ◽  
Central Difference Method ◽  
American Put

We develop an accurate finite difference scheme for pricing two-asset American put options. We use the central difference method for space derivatives and the implicit Euler method for the time derivative. Under certain mesh step size limitations, the matrix associated with the discrete operator is an M-matrix, which ensures that the solutions are oscillation-free. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. It is shown that the scheme is second-order convergent with respect to the spatial variables. Numerical results support the theoretical results.

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