Approximation by pseudo-linear operators

  • Authors:

  • Barnabás Bede;Hajime Nobuhara;Martina Daňková;Antonio Di Nola

  • Affiliations:

  • Department of Mathematics, The University of Texas- Pan American, Edinburg, TX 78541, USA;Department of Intelligent Interaction Technologies, Graduate School of Systems and Information Engineering, University of Tsukuba, Tenoudai 1-1-1, Tsukuba science city, Ibaraki 305-8573, Japan;Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. Dubna 22, 701 03 Ostrava, Czech Republic;Dipartamento di Matematica e Informatica, Universitá di Salerno, Via S. Allende, 84081 Baronissi (Salerno), Italy

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2008

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Abstract

The approximation operators provided by classical approximation theory use exclusively as underlying algebraic structure the linear structure of the reals. Also they are all linear operators. We address in the present paper the following problems: Need all the approximation operators be linear? Is the linear structure the only one which allows us to construct particular approximation operators? As an answer to this problem we propose new, particular, pseudo-linear approximation operators, which are defined in some ordered semirings. We study these approximations from a theoretical point of view and we obtain that these operators have very similar properties to those provided by classical approximation theory. In this sense we obtain uniform approximation theorems of Weierstrass type, and Jackson-type error estimates in approximation by these operators.