Approximation by truncated max‐product operators of Kantorovich‐type based on generalized (ϕ,ψ)‐kernels

Abstract

Suggested by the max‐product sampling operators based on sinc‐Fejér kernels, in this paper, we introduce truncated max‐product Kantorovich operators based on generalized type kernels depending on two functions ϕ and ψ satisfying a set of suitable conditions. Pointwise convergence, quantitative uniform convergence in terms of the moduli of continuity, and quantitative Lp‐approximation results in terms of a K‐functional are obtained. Previous results in sampling and neural network approximation are recaptured, and new results for many concrete examples are obtained.

A Quantitative Estimate for the Sampling Kantorovich Series in Terms of the Modulus of Continuity in Orlicz Spaces

“…For the sake of completeness, we recall that the well-known (above mentioned) sinc-function is that defined as sin(πx)/πx, if x = 0, and 1 if x = 0, see e.g., [26,27]. For other examples of kernels, see, e.g., [13,20,15,22,16].…”

Convergence of sampling Kantorovich operators in modular spaces with applications

“…where f ∶ ℝ → ℝ is any locally integrable function for which the above series is convergent for every x ∈ ℝ , (t k ) k∈ℤ is a suitable sequence of real numbers, with 0 < Δ k ∶= t k+1 − t k , k ∈ ℤ , and ∶ ℝ → ℝ is a kernel function satisfying suitable assumptions, w > 0 see, e.g., [6,18,32,33,50,60]. The operators in (I) represent the L p -version of the generalized sampling operators (see, e.g., [9, 10, 13, 17, 25, 28-30, 51, 61]) introduced and studied by P.L.…”

Approximation by pseudo-linear discrete operators

“…On the other hand, Bede et al (see [13]) defined a new type of nonlinear operator by using three types of ordered semirings. Maxproduct and max-min operators, two of these operators, are frequently studied in approximation and fuzzy theory (see [11,21,22,23,24,25,26,27,29,30,31,32,33,34,35,36,37,39,41,43]), although the approximation properties of generated pseudo-linear operators are examined a few in the literature (see [12,13,14]). It should also observe that the generated pseudo-linear operators may perform better results (see [14]) than the max-product, max-min and linear counterparts.…”