On approximation properties of a family of linear operators at critical value of parameter

Abstract

We introduce the family of linear operatorsassociated to a certain "admissible bunch" of operators S t , t > 0, acting on L p (R n , dm, and investigate the approximation properties of this family as → 0 + . We give some applications to the Riesz and the Bessel potentials generated by the ordinary (Euclidean) and generalized translations. .tr (I.A. Aliev), frteb@aas.ab.az (A.D. Gadjiev), aral@science.ankara.edu.tr (A. Aral).

On Certain Multidimensional Nonlinear Integrals

“…Using integration by parts for Stieltjes integrals and relation (2.2), we get the following inequality: 1) ! ( r; 0)dr:…”

“…Some studies on nonlinear operators in di¤erent settings can be found in [5,9,13,21]. Also, results and applications in wide range concerning linear operators can be found in [1,4,8,12,17,20]. Some weighted approximation results concerning well-known Gauss-Weierstrass and Picard integral operators can be found in the recent articles [23] and [24], respectively.…”

Approximation by complex potentials generated by the Gamma function

“…But there exists α 0 ∈ (0, 1] , such that if C is an absolute constant with 0 < C < ln(2) − 1 2 , then we have 2)…”

“…In the real case, the approximation properties of the potentials such as those of Riesz, Bessel, generalized Riesz, generalized Bessel and Flett have been studied by many authors, see e.g. Kurokawa [5], Gadjiev-Aral-Aliev [3], Uyhan-Gadjiev-Aliev [7], Sezer [6], Aliev-Gadjiev-Aral [1] and their references.…”

Two Forms of An Inverse Operator to the Generalized Bessel Potential

“…Further investigation of the hypersingular integrals associated to G α was performed in [13]. In [14], the fractional powers…”