Poisson Cauchy Research Articles

Approximation by multivariate generalized Poisson–Cauchy type singular integral operators

This research and survey work deals exclusively with the study of the approximation of generalized multivariate Poisson–Cauchy type singular integrals to the identity-unit operator. Here we study quantitatively most of their approximation properties. These operators are not in general positive linear operators. In particular we study the rate of convergence of these integral operators to the unit operator, as well as the related simultaneous approximation. These are given via Jackson type inequalities and by the use of multivariate high order modulus of smoothness of the high order partial derivatives of the involved function. Also we study the global smoothness preservation properties of these integral operators. These multivariate inequalities are nearly sharp and in one case the inequality is attained, that is sharp. Furthermore we give asymptotic expansions of Voronovskaya type for the error of approximation. The above properties are studied with respect to $$L_{p}$$ norm, $$1\le p\le \infty $$ .

The compound truncated Poisson Cauchy model: A descriptor for multimodal data

Multimodal data are often present in synthetic aperture radar (SAR) image processing. Such images are often modeled by probability mixtures, but such solution may involve a large number of parameters and its inference becomes challenging. To address this issue, we proposed a probability distribution capable of describing multimodal data with only three parameters. The introduced model is defined by the sum of a random number, following the truncated Poisson law, of independent random variables with the Cauchy model, called compound truncated Poisson Cauchy (CTPC) distribution. We derived the characteristic function (cf), a distance measure between cfs, and provided estimators for the CTPC parameters: maximum likelihood estimators (MLEs) and quadratic distance estimators (QDEs). Furthermore, we derived a goodness-of-fit (GoF) distance based on the CTPC law and its empirical cf. To quantify the performance of the proposed estimators and GoF statistic, we employed a Monte Carlo simulation and the results suggest that QDEs outperform MLEs. Finally, we detail a numerical experiment with actual SAR data which describes a segment of SAR intensities with at least two types of textures. Our model could outperform six SAR intensities models: Weibull, gamma, generalized gamma, K, G0, and beta generalized normal.

Correction to: Quantitative Approximation by Nonlinear Picard–Choquet, Gauss–Weierstrass–Choquet and Poisson–Cauchy–Choquet Singular Integrals

By using the concept of Choquet nonlinear integral with respect to a submodular set function, we introduce the nonlinear Picard–Choquet operators, Gauss–Weierstrass–Choquet operators and Poisson–Cauchy–Choquet operators with respect to a family of submodular set functions. Quantitative approximation results of the order $$\omega _{1}(f; t)_{\mathbb {R}}$$, $$t>0$$, are obtained with respect to the Choquet measure $$\mu (A)=\sqrt{M(A)}$$ where M represents the Lebesgue measure, and with respect to some families of possibility measures. Also, due to the possibilities of choice for the submodular set functions, for some subclasses of functions we prove that these Choquet type operators have essentially better approximation properties than their classical correspondents.

Quantitative Approximation by Nonlinear Picard–Choquet, Gauss–Weierstrass–Choquet and Poisson–Cauchy–Choquet Singular Integrals

Quantitative Approximation by Nonlinear Picard–Choquet, Gauss–Weierstrass–Choquet and Poisson–Cauchy–Choquet Singular Integrals

Feller’s Scheme in Approximation by Nonlinear Possibilistic Integral Operators

ABSTRACTBy analogy with Feller’s general probabilistic scheme used in the construction of many classical convergent sequences of linear operators, in this paper, we consider a Feller-kind scheme based on the possibilistic integral, for the construction of convergent sequences of nonlinear operators. In particular, in the discrete case, all the so-called max-product Bernstein-type operators and their qualitative convergence properties are recovered. Also, discrete nonperiodic nonlinear possibilistic convergent operators of Picard type, Gauss–Weierstrass type and Poisson–Cauchy type are studied and the possibility of introduction of discrete periodic(trigonometric) nonlinear possibilistic operators of de la Vallée–Poussin type, of Fejér type and of Jackson type is mentioned as future directions of research.

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.

Lp -general approximations by multivariate singular integral operators

Abstract In this article, we study the Lp , 1 ≤ p < ∞ approximation properties of general multivariate singular integral operators over RN , N ≥ 1. We establish their convergence to the unit operator with rates. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss–Weierstrass, Poisson Cauchy and trigonometric singular integral operators where this theory can be applied directly.

Statistical Approximation by Double Poisson–Cauchy Singular Integral Operators

In this paper we show that it is possible to approximate a continuous and 2π-periodic function on the disk centered at origin with radius π by means of double Poisson–Cauchy singular integral operators which do not need to be positive in general. Our results cover not only the classical approximation but also the statistical approximation process.

Global smoothness preservation and simultaneous approximation for multivariate general singular integral operators

In this article we continue with the study of multivariate smooth general singular integral operators over R N , N ≥ 1 , regarding their simultaneous global smoothness preservation property with respect to the L p norm, 1 ≤ p ≤ ∞ , by involving multivariate higher order moduli of smoothness. Also we study their multivariate simultaneous approximation to the unit operator with rates. The multivariate Jackson type inequalities obtained are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function. In the uniform case of global smoothness we prove optimality. At the end we list the multivariate Picard, Gauss–Weierstrass, Poisson–Cauchy and Trigonometric singular integral operators as applicators of our general theory.

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] .

A Voronovskaya Type Theorem for Poisson–Cauchy Type singular operators

In this article we continue with the study of approximation properties of smooth Poisson–Cauchy Type singular integral operators over the real line. We produce Voronovskaya Type results and give some quantitative results regarding the rate of convergence of these singular integral operators.

Global smoothness and uniform convergence of smooth Poisson–Cauchy type singular operators

In this article we introduce the smooth Poisson–Cauchy type singular integral operators over the real line. Here we study their simultaneous global smoothness preservation property with respect to the L p norm, 1 ⩽ p ⩽ ∞ , by involving higher order moduli of smoothness. Also we study their simultaneous approximation to the unit operator with rates involving the modulus of continuity with respect to the uniform norm. The produced Jackson type inequalities are almost sharp containing elegant constants, and they reflect the high order of differentiability of the engaged function.

General Theory of Global Smoothness Preservation by Singular Integrals, Univariate Case

In this article it is established that the well-known singular integrals of Picard, Poisson–Cauchy, Gauss–Weierstrass, and their Jackson-type generalizations fulfill the “global smoothness preservation” property, i.e., they “ripple” less than the function they are applied on, that is, producing a nice and fit approximation to the unit. The related results are established over various spaces of functions and the associated inequalities involve different types of corresponding moduli of smoothness. Several times these inequalities are proved to be sharp, namely, they are attained.