A subsequence principle is obtained, characterizing Banach spacescontaining co' in the spirit of the author's 1974 characterization of Banachspaces containing II . Defiaition. A sequence(bj )in a Banach space is called strongly summing (s.s.)if (bj ) is a weak-Cauchy basic sequence so that whenever scalars (cj ) satisfysUPn II E~=I cijll < 00, thenECj converges.A...

In this paper, we extend the study of statistical convergence complex uncertain sequences in a given uncertainty space. We produce relation between and an environment. also initiate statistically Cauchy sequence to prove that is convergent if only it Cauchy. further characterize via boundedness density operator.

For an entire function $f(z) = \sum_{k=0}^\infty a_k z^k,$ $a_k >0,$ we define the sequence of second quotients Taylor coefficients $Q := \left( \frac{a_k^2}{a_{k-1}a_{k+1}} \right)_{k=1}^\infty$. We find new necessary conditions for a with non-decreasing $Q$ to belong Laguerre--Polya class type I. also estimate possible number nonreal zeros $Q.$

Let {Zk}k ≥ 1 denote a sequence of independent Bernoulli random variables defined by P(Zk = 1) 1/k 1− 0) (k and put Tn ≔ ∑1 ≤ k n kZk. It is known that Tn/n convergesweakly to real variable D with density proportional the Dickman function, delay-differential equation uϱ ′ (u) + ϱ(u − 0 (u > initial condition ϱ(u) 1(0 u 1). Improving on earlier work, we propose asymptotic formulae remainders for...

A subsequence principle is obtained, characterizing Banach spacescontaining co' in the spirit of the author's 1974 characterization of Banachspaces containing II . Defiaition. A sequence(bj )in a Banach space is called strongly summing (s.s.)if (bj ) is a weak-Cauchy basic sequence so that whenever scalars (cj ) satisfysUPn II E~=I cijll < 00, thenECj converges.A...

We explore two generalizations of the Cauchy-Schwarz Bessel’s inequality and the Selberg inequality and their application to probability theory. We then give a tautological proof of the De Caen-Selberg Inequality and a proof of the second Borel-Cantelli Lemma with negative dependence. We finish with a suggestion of how linear operator theory can help us understand the tightness of many Selberg-...

In this paper we obtain bounds showing increasing stability of the continuation for solutions of the stationary Maxwell system when the wave number k is growing. We use a reduction of this system to a new system with the Helmholtz operator in the principal part and use hyperbolic energy and Carleman estimates with k-independent constants in the Cauchy problem for this equations. Hyperbolic ener...

Consider the Cauchy principal value integral I(kf;X)=i k{x)F^Ldx, -1<A<1. 7-1 x — A If we approximate f(x) by Yli=oajP¡(x'i w) where {p } is a sequence of orthonormal polynomials with respect to an admissible weight function w and û, = (/. P.), then an approximation to I(kf; X) is given by X!/=o ajl(kp¡ ; ^)If, in turn, we approximate a¡ by ajm = £TM , wimf(x¡m)Pj(xim), then we get a double seq...

A difficult step in the derivation of the strong forms of the Cauchy theorem, Green's lemma, and related theorems from the corresponding weak forms is the construction, for a given rectifiable Jordan curve J, of a sequence of Jordan polygons lying interior to / , converging to J", and having uniformly bounded lengths. This note presents what the author believes to be a simpler elementary constr...

and Applied Analysis 3 Definition 1.4. A sequence {xk} in an n-normed space X is said to converge to some x ∈ X in the n-norm if lim k→∞ ∥ ∥xk − x, y2, . . . , yn ∥ ∥ 0, 1.4 for every y2, . . . , yn ∈ X. Definition 1.5. A sequence {xk} in an n-normed space X is said to be a Cauchy sequence with respect to the n-norm if lim k,l→∞ ∥ ∥xk − xl, y2, . . . , yn ∥ ∥ 0, 1.5 for every y2, . . . , yn ∈ X...