Rate of growth and convergence factors for power methods of limitation

1974 ◽  
Vol 76 (1) ◽  
pp. 241-246 ◽  
Author(s):  
Abraham Ziv
Keyword(s):  
Radius Of Convergence ◽  
Complex Numbers ◽  
Finite Limit ◽  
Power Method ◽  
Real Constant ◽  
Rate Of Growth ◽  
Image Position

Let , where pk are complex numbers, have 0 < ρ ≤ ∞ for radius of convergence and assume that P(x) ≠ 0 for α ≤ x < ρ (α < ρ is some real constant). Assuming that is convergent for all (x ∈ [0, ρ), we define the P-limit of the sequence s = {sk} byThis, so called, power method of limitation (see (3), Definition 9 and (1) Definition 6) will be denoted by P. The best known power methods are Abel's (P(x) = 1/(1 – x), α = 0, ρ = 1) and Borel's (P(x) = ex, α = 0, ρ = ∞). By Cp we denote the set of all sequences, P-limitable to a finite limit and by the set of all sequences, P-limitable to zero.

On methods of summability based on integral functions

1959 ◽  
Vol 55 (1) ◽  
pp. 23-30 ◽  
Author(s):  
D. Borwein
Keyword(s):  
Integral Function ◽  
Radius Of Convergence ◽  
Complex Numbers ◽  
Arbitrary Complex ◽  
Image Position

Suppose throughout thatand thatis an integral function. Suppose also that l, sn(n = 0,1,…) are arbitrary complex numbers and denote by ρ(ps) the radius of convergence of the series

On a singular eigenvalue problem

1968 ◽  
Vol 64 (2) ◽  
pp. 439-446 ◽  
Author(s):  
D. Naylor ◽  
S. C. R. Dennis
Keyword(s):  
Differential Equation ◽  
Eigenvalue Problem ◽  
Bessel Functions ◽  
Asymptotic Condition ◽  
Real Constant ◽  
Limit Circle ◽  
Expansion In Eigenfunctions ◽  
Image Position

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equationHere ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equationon the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

A note on the strong regularity of Nrlund means

1983 ◽  
Vol 94 (2) ◽  
pp. 261-263
Author(s):  
J. R. Nurcombe
Keyword(s):  
Strong Convergence ◽  
Complex Numbers ◽  
Strong Regularity ◽  
Image Position

Let (pn), (qn) and (un) be sequences of real or complex numbers withThe sequence (sn) is strongly generalized Nrlund summable with index 0, to s, or s or snsN, p, Q ifand pnv=pnvpnv1, with p10. Strong Nrlund summability N, p was first studied by Borweing and Cass (1), and its generalization N, p, Q by Thorp (6). We shall say that (sn) is strongly generalized convergent of index 0, to s, and write snsC, 0, Q if sns and where sn=a0+a1++an. When qn all n, this definition reduces to strong convergence of index , introduced by Hyslop (4). If as n, the sequence (sn) is summable (, q) to s sns(, q).

Unbounded Positive Definite Functions

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart
Keyword(s):  
Abelian Group ◽  
Positive Measure ◽  
Positive Definite Function ◽  
Positive Definite ◽  
Locally Compact Abelian Group ◽  
Definite Function ◽  
Complex Numbers ◽  
Stieltjes Transform ◽  
Real Numbers ◽  
Image Position

Let G be an abelian group, written additively. A complexvalued function ƒ, defined on G, is said to be positive definite if the inequality1holds for every choice of complex numbers C1, …, cn and S1, …, sn in G. It follows directly from (1) that every positive definite function is bounded. Weil (9, p. 122) and Raïkov (5) proved that every continuous positive definite function on a locally compact abelian group is the Fourier-Stieltjes transform of a bounded positive measure, thus generalizing theorems of Herglotz (4) (G = Z, the integers) and Bochner (1) (G = R, the real numbers).If ƒ is a continuous function, then condition (1) is equivalent to the condition that2

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

Nonzero Symmetry Classes of Smallest Dimension

1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim
Keyword(s):  
Vector Space ◽  
Symmetric Group ◽  
Linear Mapping ◽  
Complex Numbers ◽  
Dimensional Vector ◽  
Complex Character ◽  
Tensor Power ◽  
Dimensional Vector Space ◽  
Symmetry Classes ◽  
Image Position

Let U be a k-dimensional vector space over the complex numbers. Let ⊗m U denote the mth tensor power of U where m ≧ 2. For each permutation σ in the symmetric group Sm, there exists a linear mapping P(σ) on ⊗mU such thatfor all x1, …, xm in U.Let G be a subgroup of Sm and λ an irreducible (complex) character on G. The symmetrizeris a projection of ⊗ mU. Its range is denoted by Uλm(G) or simply Uλ(G) and is called the symmetry class of tensors corresponding to G and λ.

scholarly journals On Morita's p-adic gamma function

1981 ◽  
Vol 89 (1) ◽  
pp. 23-27 ◽  
Author(s):  
Daniel Barsky
Keyword(s):  
Continuous Function ◽  
Prime Number ◽  
Gamma Function ◽  
Radius Of Convergence ◽  
Image Position

Y. Morita proved that, for each prime number p, one can define a p-adic continuous function Γp(x) from p to p, interpolating the sequencewhere m runs through the integers m prime to p with 1 ≤ m < n. Our aim is to show how this result is related to Dwork's result on the radius of convergence of

scholarly journals On (J, pn)-summability of Fourier series

1972 ◽  
Vol 18 (1) ◽  
pp. 13-17
Author(s):  
F. M. Khan
Keyword(s):  
Fourier Series ◽  
Power Series ◽  
Radius Of Convergence ◽  
Partial Sums ◽  
Summability Of Fourier Series ◽  
Image Position

Let pn>0 be such that pn diverges, and the radius of convergence of the power seriesis 1. Given any series σan with partial sums sn, we shall use the notationand

On the Representation of Integers as Sums of Distinct Terms from a Fixed Sequence

1966 ◽  
Vol 18 ◽  
pp. 643-655 ◽  
Author(s):  
Jon Folkman
Keyword(s):  
Fixed Sequence ◽  
Rate Of Growth ◽  
Representation Of Integers ◽  
Image Position ◽  
Positive Integers

Let A = (a1, a2, a3, …) be a sequence of positive integers. We letdenote the set of integers that are sums of distinct terms of A. If P(A) contains all sufficiently large integers, we say that A is complete. We shall show that certain classes of sequences that are characterized by their rate of growth are complete.

On a Class of Positive Linear Operators

1973 ◽  
Vol 16 (4) ◽  
pp. 557-559 ◽  
Author(s):  
J. Swetits ◽  
B. Wood
Keyword(s):  
Linear Operators ◽  
Positive Linear Operators ◽  
Complex Numbers ◽  
Image Position

In a recent paper [3] Meir and Sharma introduced a generalization of the Sα- method of summability. The elements of their matrix, (ank), are defined by(1)where is a sequence of complex numbers. if 0 < αj < l for each j = 0, 1, 2,… then ank≥0 for each n = 0, 1, 2,… and k = 0,1,2,…

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