Approximation properties of mixed sampling-Kantorovich operators

AbstractIn 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

Minimax estimation of smooth optimal transport maps

Brenier's theorem is a cornerstone of optimal transport that guarantees the existence of an optimal transport map $T$ between two probability distributions $P$ and $Q$ over $\mathbb{R}^d$ under certain regularity conditions. The main goal of this work is to establish the minimax estimation rates for such a transport map from data sampled from $P$ and $Q$ under additional smoothness assumptions on $T$. To achieve this goal, we develop an estimator based on the minimization of an empirical version of the semi-dual optimal transport problem, restricted to truncated wavelet expansions. This estimator is shown to achieve near minimax optimality using new stability arguments for the semi-dual and a complementary minimax lower bound. Furthermore, we provide numerical experiments on synthetic data supporting our theoretical findings and highlighting the practical benefits of smoothness regularization. These are the first minimax estimation rates for transport maps in general dimension.Comment: 53 pages, 6 figure

Simultaneous approximation by operators of exponential type

There are many results on the simultaneous approximation by sequences of special positive linear operators. In the year 1978, Ismail and May as well as Volkov independently studied operators of exponential type covering the most classical approximation operators. In this paper we study asymptotic properties of these class of operators. We prove that under certain conditions, asymptotic expansions for sequences of operators belonging to a slightly larger class of operators, can be differentiated term-by-term. This general theorem contains several results which were previously obtained by several authors for concrete operators. One corollary states, that the complete asymptotic expansion for the Bernstein polynomials can be differentiated term-by-term. This implies a well-known result on the Voronovskaja formula obtained by Floater

On a Durrmeyer-type modification of the Exponential sampling series

AbstractIn this paper we introduce the exponential sampling Durrmeyer series. We discuss pointwise and uniform convergence properties and an asymptotic formula of Voronovskaja type. Quantitative results are given, using the usual modulus of continuity for uniformly continuous functions. Some examples are also described

On the convergence properties of Durrmeyer-Sampling Type Operators in Orlicz spaces

Here we provide a unifying treatment of the convergence of a general form of sampling type operators, given by the so-called Durrmeyer sampling type series. In particular we provide a pointwise and uniform convergence theorem on $\mathbb{R}$, and in this context we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Then we obtain a modular convergence theorem in the general setting of Orlicz spaces $L^\varphi(\mathbb{R})$. From the latter result, the convergence in $L^p(\mathbb{R})$-space, $L^\alpha\log^\beta L$, and the exponential spaces follow as particular cases. Finally, applications and examples with graphical representations are given for several sampling series with special kernels