L^{p}-Approximation and Shape-preserving Properties of the Max-product Generalized Sampling Operators

Abstract

In this paper, we investigate the convergence in the $L^{p}$-norm and certain shape-preserving properties of the max-product generalized sampling operators. More precisely, we establish quantitative estimates for the approximation error in the $L^{p}$-norm, for $ 1 \le p < +\infty$, in the case of non-negative and bounded functions defined on $[-1,1]$. These estimates are derived by means of the so-called $τ$-modulus, an averaged modulus of smoothness introduced by Sendov and Popov. As a direct consequence, we prove that the max-product generalized sampling operators $L^{p}$-converge to non-negative functions that are measurable, bounded and Riemann integrable on the interval $[-1,1]$. In the final section, we extend several shape-preserving results of Coroianu and Gal, originally established for specific kernels (such as the sinc/Whittaker and Fejér kernels), to the broader class of smooth centered bell-shaped kernels. Under suitable assumptions on the kernel, we prove that the max-product generalized sampling operators partially preserve the monotonicity of any function $f:[0,1] \rightarrow \R_{0}^{+}$ that is either non-decreasing or non-increasing on $[0,1]$.