A note on the strong regularity of Nrlund means

1983 ◽  
Vol 94 (2) ◽  
pp. 261-263
Author(s):  
J. R. Nurcombe

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).

1969 ◽  
Vol 21 ◽  
pp. 1309-1318 ◽  
Author(s):  
James Stewart

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


Author(s):  
V. Krishna Kumar

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, ∞).


1980 ◽  
Vol 32 (4) ◽  
pp. 957-968 ◽  
Author(s):  
G. H. Chan ◽  
M. H. Lim

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.


1973 ◽  
Vol 16 (4) ◽  
pp. 557-559 ◽  
Author(s):  
J. Swetits ◽  
B. Wood

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,…


1969 ◽  
Vol 65 (2) ◽  
pp. 461-465
Author(s):  
R. N. Jain

MacRobert (4–7) and Ragab(8) have summed many infinite and finite series of E-functions by expressing the E-functions as Barnes integrals and interchanging the order of summation and integration. Verma (9) has given two general expansions involving E-functions from which, in addition to some new results, all the expansions given by MacRobert and Ragab can be deduced. Proceeding similarly, we have studied general summations involving H-functions. The H-function is the most generalized form of the hypergeometric function. It contains a vast number of well-known analytic functions as special cases and also an important class of symmetrical Fourier kernels of a very general nature. The H-function is defined as (2)where 0 ≤ n ≤ p, 1 ≤ m ≤ q, αj, βj are positive numbers and aj, bj may be complex numbers.


1965 ◽  
Vol 5 (1) ◽  
pp. 1-7 ◽  
Author(s):  
B. Kwee

Let {Pn} be any sequence of real or complex numbers subject to the sole restriction And let If tn → s, whenever sn → s we say that the sequence {sn} is summable Nörlund or summable (N, p) to s.


1974 ◽  
Vol 76 (1) ◽  
pp. 241-246 ◽  
Author(s):  
Abraham Ziv

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.


Author(s):  
J. BOURGAIN ◽  
M. Z. GARAEV

AbstractLet Fp be the field of a prime order p and F*p be its multiplicative subgroup. In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 < |A| < p12/23. Then we apply our estimate to obtain explicit bounds for some exponential sums in Fp. We show that for any subsets X, Y, Z ⊂ F*p and any complex numbers αx, βy, γz with |αx| ≤ 1, |βy| ≤ 1, |γz| ≤ 1, the following bound holds: We apply this bound further to show that if H is a subgroup of F*p with |H| > p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than $M^{3-1/24+o(1)}\Bigl(1+(M^2/p)^{1/24}\Bigr).$. This implies that if p1/2 < M < p, then


1968 ◽  
Vol 8 (1) ◽  
pp. 109-113 ◽  
Author(s):  
A. Ran

Let Ω be the group of the functions ƒ(z) of the complex variable z, analytic in some neighborhood of z = 0, with ƒ(0) = 0, ƒ′(0) = 1, where the group operation is the composition g[f(z)](g(z), f(z) ∈ Ω). For every function f(z) ∈ Ω there exists [4] a unique formal power series where the coefficients ƒq(s) are polynomials of the complex parameter s, with ƒ1(s) = 1, such that and, for any two complex numbers s and t, the formal law of composition is valid.


1971 ◽  
Vol 70 (2) ◽  
pp. 257-262
Author(s):  
Dale W. Swann

Let K(s, t) be a complex-valued L2 kernel on the square ⋜ s, t ⋜ by which we meanand let {λν}, perhaps empty, be the set of finite characteristic values (f.c.v.) of K(s, t), i.e. complex numbers with which there are associated non-trivial L2 functions øν(s) satisfyingFor such kernels, the iterated kernels,are well-defined (1), as are the higher order tracesCarleman(2) showed that the f.c.v. of K are the zeros of the modified Fredhoim determinantthe latter expression holding only for |λ| sufficiently small (3). The δn in (3) may be calculated, at least in theory, by well-known formulae involving the higher order traces (1). For our later analysis of this case, we define and , respectively, as the minimum and maximum moduli of the zeros of , the nth section of D*(K, λ).


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