Cauchy Singular Operator Research Articles

Lp approximation with rates by multivariate generalized discrete singular operators

Here we give the approximation properties with rates of multivariate generalized discrete versions of Picard, Gauss–Weierstrass, and Poisson–Cauchy singular operators over RN,N ≥ 1. We treat both the unitary and non-unitary cases of the operators above. We derive quantitatively Lp convergence of these operators to the unit operator by involving the Lp higher modulus of smoothness of an Lp function.

On the factorization of bounded measurable functions in the classes of Cauchy type integrals with density from Lp(·)(Γ;ω)

Let Γ be a simple rectifiable closed Carleson curve and be its measurable bounded nonzero function on Γ. It is proved in the paper that the factorization of such a function in the class of Cauchy type integrals with density from the Lebesgue space Lp(·)(Γ;ω) with variable exponent is equivalent to the property of the singular integral operator , where is the operator generated by a Cauchy singular operator, to be Noetherian in the space Lp(·)(Γ;ω). This result generalizes a theorem of I. B. Simonenko.

Quantitative approximation by fractional smooth Poisson Cauchy singular operators

In this article we study the very general fractional smooth Poisson Cauchy singular integral operators on the real line, regarding their convergence to the unit operator with fractional rates in the uniform norm. The related established inequalities involve the higher order moduli of smoothness of the associated right and left Caputo fractional derivatives of the engaged function. Furthermore, we produce a fractional Voronovskaya type of result giving the fractional asymptotic expansion of the basic error of our approximation. We finish with applications. Our operators are not in general positive. We are mainly motivated by Anastassiou (submitted for publication) [1] .

Rational approximation in Orlicz spaces on Carleson curves

AbstractWe define some subclasses of Orlicz spaces of functions and establish herea direct theorem of the approximation theory by rational functions. 1 Introduction and main results Let Γ be a rectifiable Jordan curve in the complex plane C and let G:= IntΓ,G − := ExtΓ. Without loss of generality we suppose that 0 ∈ G.Further let T :={w∈ C : |w| = 1}, U:= IntT,U − := ExtT.We denote by ϕand ϕ 1 the conformalmappings of G − and Gonto U − normalized by the conditionsϕ(∞) = ∞, lim z→∞ ϕ(z)z>0andϕ 1 (0) = ∞, lim z→0 zϕ 1 (z) >0respectively and let ψand ψ 1 be the inverse mappings of ϕand ϕ 1 .Let also L p (Γ) and E p (G) (1 ≤ p<∞) be the Lebesgue space of measurablecomplex valued functions on Γ and the Smirnov class of analytic functions in Grespectively. Since Γ is rectifiable, we have ϕ 0 ∈ E 1 (G − ), ϕ 0 1 ∈ E 1 (G) and ψ 0 , Received by the editors June 2003.Communicated by R. Delanghe.1991 Mathematics Subject Classification : Primary 30E10, 41A10, 41A20. Secondary 41A25,46E30.Key words and phrases : Carleson curve, Cauchy singular operator, Faber polynomials, Orliczspace, Smirnov-Orlicz class.Bull. Belg. Math. Soc. 12(2005), 223–234

A criterion for the weighted Cauchy singular operator to belong to the algebra of singular integral operators with coefficients in a Sarason algebra

A criterion for the weighted Cauchy singular operator to belong to the algebra of singular integral operators with coefficients in a Sarason algebra

On the boundedness of Cauchy singular operator from the spaceL p toL q ,p&gt;q&gt;-1

It is proved that for a Cauchy type singular operator, given by equality (1), to be bounded from the Lebesgue space Lp(€) to Lq(€), as € = ( 1=1 €n, €n = fz : jzj = rng, it is necessary and sucient that either condition (4) or (5) be fulÞlled.