244 research outputs found
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 between two probability distributions
and over 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 and under additional smoothness
assumptions on . 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
, 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 . From the latter result, the
convergence in -space, , and the
exponential spaces follow as particular cases. Finally, applications and
examples with graphical representations are given for several sampling series
with special kernels
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