Approximation properties of mixed sampling-Kantorovich operators
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Approximation properties of mixed sampling-Kantorovich operators Laura Angeloni1
· Danilo Costarelli1 · Gianluca Vinti1
Received: 24 February 2020 / Accepted: 17 September 2020 © The Author(s) 2020
Abstract In the present paper we study the pointwise and uniform convergence properties of a family of multidimensional sampling Kantorovich type operators. Moreover, besides convergence, quantitative estimates and a Voronovskaja type theorem have been established. Keywords Sampling-Kantorovich operators · Pointwise convergence · Uniform convergence · Order of approximation · Asymptotic expansions · Voronovskaja formulae Mathematics Subject Classification 41A30 · 41A05
1 Introduction In the last years there was an increasing interest in approximation by means of families of discrete operators in several function spaces both in the one-dimensional case and in multidimensional setting, also thanks to the applicative outcome of such results (see, e.g., [18,27,28,36,37,41]). For example, the generalized sampling series defined as k (Sw f )(t) = f χ(wt − k), () w N k∈Z
RN ,
t ∈ w > 0, where χ is a kernel, have been widely studied with respect to several notions of convergence, such as pointwise, uniform, L p , modular convergence [23–25] and also, recently [5,7], convergence in variation (for other approximation results in BV-spaces see, e.g., [9–13]). The interest of such operators is also due to their deep connections with problems of Signal and Image Processing: indeed they furnish an approximate version of
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Laura Angeloni [email protected] Danilo Costarelli [email protected] Gianluca Vinti [email protected]
1
Department of Mathematics and Computer Science, University of Perugia, Via Vanvitelli 1, 06123 Perugia, Italy 0123456789().: V,-vol
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the Shannon Sampling Theorem (see, e.g., [46,47]) that allows to reconstruct not necessarily band-limited signals (images). In order to obtain convergence in variation for the operators (), in [7] a new family of discrete operators has been introduced: ⎡ ⎤ k j +1 w k k 1 N ⎣w (K w, j f )(t) := f ,...,u ..., du ⎦ χ(wt − k), (∗) kj w w N w k∈Z
RN , w
> 0 and j = 1, . . . , N . The operators (∗), that here will be called mixed for every t ∈ sampling-Kantorovich operators, are essentially a Kantorovich version of the generalized sampling series where the integral mean is computed on just one section
of the function involved, while the usual Kantorovich version replaces the whole value f wk with an integral mean on a multidimensional interval around the sampling node wk , i.e., N f (u) du χ(wt − k), (∗∗) (K w f )(t) := w k k +1 k∈Z N
j N j=1 [ w
,
j w
]
for every t ∈ R N , w > 0. The introduction of the operators (∗) was naturally motivated by the fact that such operators allow to obtain a multidimensional generalization of the classical relation proved by Lorentz among the derivative of the Bernstein polynomials and the Kantorovich polynomials acting on the derivative of the function, i
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