Approximation Properties of the Sampling Kantorovich Operators: Regularization, Saturation, Inverse Results and Favard Classes in $$L^p$$-Spaces

Abstract

AbstractIn the present paper, a characterization of the Favard classes for the sampling Kantorovich operators based upon bandlimited kernels has been established. In order to achieve the above result, a wide preliminary study has been necessary. First, suitable high order asymptotic type theorems in$$L^p$$Lp-setting,$$1 \le p \le +\infty $$1≤p≤+∞, have been proved. Then, the regularization properties of the sampling Kantorovich operators have been investigated. Here, we show how the regularity of the kernel influences the operator itself; this has been shown for bandlimited kernels, or more in general for kernels in Sobolev spaces, satisfying a Strang-Fix type condition of order$$r \in \mathbb {N}^+$$r∈N+. Further, for the order of approximation of the sampling Kantorovich operators, quantitative estimates based on the$$L^p$$Lpmodulus of smoothness of orderrhave been established. As a consequence, the qualitative order of approximation is also derived assumingfin suitable Lipschitz and generalized Lipschitz classes. Moreover, an inverse theorem of approximation has been stated, allowing to obtain a full characterization of the Lipschitz and of the generalized Lipschitz classes in terms of convergence of the above sampling type series. These approximation results have been proved for not necessarily bandlimited kernels. From the above mentioned characterization, it remains uncovered the saturation case that, however, can be treated by a totally different approach assuming that the kernel is bandlimited. Indeed, since sampling Kantorovich (discrete) operators based upon bandlimited kernels can be viewed as double-singular integrals, exploiting the properties of the convolution in Fourier Analysis, we become able to get the desired result obtaining a complete overview of the approximation properties in$$L^p(\mathbb {R})$$Lp(R),$$1 \le p \le +\infty $$1≤p≤+∞, for the sampling Kantorovich operators.