Razvan A. Mezei

Uniform convergence with rates of smooth Gauss–Weierstrass singular integral operators

In this article, we introduce and study the smooth Gauss–Weierstrass singular integral operators on the line of very general kind. We establish their convergence to the unit operator with rates. The estimates are mostly sharp and they are pointwise or uniform. The established inequalities involve the higher order modulus of smoothness. To prove optimality we use mainly the geometric moment theory method.

Uniform convergence with rates for smooth Poisson-Cauchy-type singular integral operators

In this article we continue the study of smooth Poisson-Cauchy-type singular integral operators on the line of very general kind. We establish their uniform convergence to the unit operator, with rates. The estimates are mostly sharp and they are pointwise or uniform. The inequalities established involve the higher order modulus of smoothness. To prove optimality we use mainly the geometric moment theory method.

UNIFORM CONVERGENCE WITH RATES OF GENERAL SINGULAR OPERATORS

George A. Anastassiou and Razvan A. MezeiThe University of MemphisDepartment of Mathematical Sciences,Memphis, TN 38152, U.S.A.ganastss@memphis.edu, rmezei@memphis.eduABSTRACTIn this article we study the approximation properties of general singular integral oper-ators over the real line. We establish their convergence to the unit operator with rates.The estimates are mostly sharp and they are pointwise or uniform. The establishedinequalities involve the higher order modulus of smoothness. We apply this theory tothe trigonometric singular operators.RESUMENEn este art´iculo estudiamos propiedades de aproximaci´on de operadores integralessingulares generales sobre la recta real. Establecemos su convergencia al operadorunidad con las tasas correspondientes. Las estimaciones son mayormente ajustadasy son tanto puntuales como uniformes. Las desigualdades encontradas involucran elm´odulo de suavidad de alto orden. Aplicamos esta teor´ia a los operadores singularestrigonom´etricos.Keywords and Phrases: Best constant, general singular integral, trigonometric singular integral,modulus of smoothness, sharp inequality.2010 AMS Mathematics Subject Classification: 26A15, 41A17, 41A35.

Quantitative approximation by fractional smooth general singular operators

Abstract In this article we study the fractional smooth general 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 result giving the fractional asymptotic expansion of the basic error of our approximation. We finish with applications to fractional trigonometric singular integral operators. Our operators are not in general positive.

Numerical Analysis Using Sage

This is the first numerical analysis text to use Sage for the implementation of algorithms and can be used in a one-semester course for undergraduates in mathematics, math education, computer science/information technology, engineering, and physical sciences. The primary aim of this text is to simplify understanding of the theories and ideas from a numerical analysis/numerical methods course via a modern programming language like Sage. Aside from the presentation of fundamental theoretical notions of numerical analysis throughout the text, each chapter concludes with several exercises that are oriented to real-world application. Answers may be verified using Sage. The presented code, written in core components of Sage, are backward compatible, i.e., easily applicable to other software systems such as Mathematica. Sage is open source software and uses Python-like syntax. Previous Python programming experience is not a requirement for the reader, though familiarity with any programming language is a plus. Moreover, the code can be written using any web browser and is therefore useful with Laptops, Tablets, iPhones, Smartphones, etc. All Sage code that is presented in the text is openly available on SpringerLink.com.

Reverse and Forward Fractional Integral Inequalities

Here we present reverse Lp fractional integral inequalities for left and right Riemann-Liouville, generalized Riemann-Liouville, Hadamard, Erdelyi-Kober and multivariate Riemann-Liouville fractional integrals. Then we derive reverse Lp fractional inequalities regarding the left Riemann-Liouville, the left and right Caputo and the left and right Canavati type fractional derivatives. We finish the article with general forward fractional integral inequalities regarding Erdelyi-Kober and multivariate Riemann-Liouville fractional integrals by involving convexity.

Applications and Lipschitz results of Approximation by Smooth Picard and Gauss-Weierstrass Type Singular Integrals

Continuamos nuestros estudios sobre convergencia uniforme de orden superior con radios y sobre convergencia Lp con radios. Concretamente, en este articulo establecemos algunos resultados de tipo Lipschitz para operadores integrales suves del tipo Picard singulares y para operadores integrales singulares de tipo Gauss-Weierstrass.

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].