A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is considered. A sufficient condition for a bounded action of this operator from a subspace to another subspace of weight Lebesgue space of functions is found. These conditions are not identical withMuckenhoupt conditions. Moreover, the completeness, minimality, and basicity of sines and cosines systems are conside...

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...

In this paper, we define several ideal versions of Cauchy sequences and completeness in quasimetric spaces. Some examples are constructed to clarify their relationships. We also show that: (1) if a quasi-metric space (X, ?) is I-sequentially complete, for each decreasing sequence {Fn} nonempty I-closed sets with diam{Fn} ? 0 as n ?, then ?n?N Fn single-point set; (2) let I be P-ideal, every pre...

1 Exercise 1 The series ∑∞ n m an converges, by de nition, exactly when the partial sum SN ∑N n m an converges as N →∞. Now using the completeness of R, this happens exactly when (SN )∞ N m is a Cauchy sequence. So it su ces to show that this is equivalent to the condition given in the exercise. This is almost trivial, but we introduce the proof anyway to please some meticulous readers. • Suppo...

We introduce and study the notions of a strongly completable and of a strongly complete quasi-uniform space. A quasi-uniform space (X,U) is said to be strongly complete if every Cauchy filter (in the sense of Sieber and Pervin) clusters in the uniform space (X,U ∨ U−1). An interesting motivation for the study of this notion of completeness is the fact, proved here, that the quasi-uniformity ind...

The space l^p with 1≤p∞ is the set of real numbers that satisfy _(n=1)^∞▒〖|x_n |^p∞〗.The function in vector X which has value fulfills norm-2 properties denoted by ,⋅‖ and pair (X,‖⋅,⋅‖) called space.A said to be complete or a Banach-2 if every Cauchy sequence converges an element space.This research was conducted prove norm-2.The first step completeness norm contained satisfies norm-2.Next, de...

and Applied Analysis 3 By (1) we have for allm ∈ N such thatm ≥ n, d (x α n , x α m ) ⪯ φ (x α n ) − φ (x α m ) . (7) Let n → ∞, by (6) we have lim m,n→∞ [φ(x α n ) − φ(x α m )] = θ and hence lim m,n→∞ d(x α n , x α m ) = θ by the normality of P. Moreover by Remark 2, {x α n } is a Cauchy sequence in X. Therefore by the completeness of X, there exists some x ∈ X such that

The purpose of this paper is to present the relations among completeness sequences, filters and nets in framework fuzzy quasi-pseudometric spaces. In particular, we show that right sequences are equivalent under special conditions quasi-pseudometrics. By introducing a kind more general K-Cauchy spaces, equivalence between sequential established.

In the present paper we study lightlike submanifolds of almost paracontact metric manifolds. We define invariant lightlike submanifolds. We study radical transversal lightlike submanifolds of para-Sasakian manifolds and investigate the geometry of distributions. Also we introduce a general notion of paracontact Cauchy-Riemann (CR) lightlike submanifolds and we derive some necessary and sufficie...