Conditions on 7c and / are given for the pointwise and uniform convergence to the Cauchy principal value integral rmm _1<A<1, j-i x~x of a sequence of integrals of piecewise linear approximations to f(x) or g\(x) = (f(x) — f(X))/(x — A). The important special case, k(x) = (1 a;)a(l + x)13, is considered in detail.

We show that several weakenings of the Cauchy condition are all equivalent under the assumption of countable choice, and investigate to what extent choice is necessary. We also show that the syntactically reminiscent notion of metastability allows similar variations, but in terms of its computational content is an empty notion. §1. Almost Cauchyness. Apart from the last section, we work in Bish...

Definition 1.3 (Banach Spaces). It is easy to show that any convergent sequence in a normed linear space is a Cauchy sequence. However, it may or may not be true in an arbitrary normed linear space that all Cauchy sequences are convergent. A normed linear space X which does have the property that all Cauchy sequences are convergent is said to be complete. A complete normed linear space is calle...

We deene a residue current of a holomorphic mapping , or more generally a holomorphic section to a holomorphic vector bundle, by means of Cauchy-Fantappie-Leray type formulas , and prove that a holomorphic function that annihilates this current belongs to the corresponding ideal locally. We also prove that the residue current coincides with the Colee-Herrera current in the case of a complete in...

A function f ∈ C(Ω) is holomorphic on Ω, if it satisfies the CauchyRiemann equations: ∂̄f = ∑n k=1 ∂f ∂z̄k dz̄k = 0 in Ω. Denote the set of holomorphic functions on Ω by H(Ω). The Bergman projection, B0, is the orthogonal projection of square-integrable functions onto H(Ω)∩L2(Ω). Since the Cauchy-Riemann operator, ∂̄ above, extends naturally to act on higher order forms, we can as well define Bergm...

We study new sequence spaces associated to sequences in normed spaces and the band matrix F̂ defined by the Fibonacci sequence. We give some characterizations of continuous linear operators and weakly unconditionally Cauchy series by means of completeness of the new sequence spaces. Also, we characterize the barreledness of a normed space via weakly∗ unconditionally Cauchy series in [Formula: se...

There are several sections. The goal is to link complex analysis and imbeddings of graphs so that graph imbeddings can be performed systematically using identification spaces based on group actions in the complex plane. We give a short primer of basic calculus and then explain some of the finer aspects that obfuscate a complete solution to the Riemann Hypothesis. This is not central to our main...

Let (X, d) be a metric space. The goal of these notes is to construct a complete metric space which contains X as a subspace and which is the “smallest” space with respect to these two properties. The resulting space will be denoted by X and will be called the completion of X with respect to d. The hard part is that we have nothing to work with except X itself, and somehow it seems we have to p...

Convergence with respect to a valuation is discussed as convergence of a Cauchy sequence. Cauchy sequences of polynomials are defined. They are used to formalize Hensel’s lemma.

The aim of this paper is to introduce the concept of convergence of a sequence on hypernormed spaces and establish a few basic properties of convergent sequences and Cauchy sequences on hypernormed spaces. Also we have established a necessary and sufficient condition for a Cauchy sequence to be convergent sequence in this spaces. In fact, also it has been shown that limit of a convergent sequen...