In this work, we develop and study an empirical projection operator scheme for solving nonparametric regression problems. This scheme is based on an approximate projection of the regression function over a suitable reproducing kernel Hilbert space (RKHS). The RKHS considered in this paper are generated by the Mercer kernels given by the Legendre Christoffel-Darboux and convolution Sinc kernels. We provide error and convergence analysis of the proposed scheme under the assumption that the regression function belongs to some suitable functional spaces. We also consider the popular RKHS regularized least square minimization for nonparametric regression. In particular, we check the numerical stability of this second scheme and we provide its convergence rate in the special case of the Sinc kernel. Finally, we illustrate the proposed methods by various numerical simulation.

翻译：在这项工作中,我们制定并研究一个解决非参数回归问题的实证预测操作员计划,这个计划的基础是对适当复制的内核Hilbert空间(RKHS)的回归功能的大致预测,本文中考虑的RKHS是由Tulturre Christoffel-Darboux和 Convolution Sinc内核提供的Mercer内核生成的。我们根据回归功能属于某些适当功能空间的假设,对拟议的计划提供错误和趋同分析。我们还考虑到流行的RKHS对非参数回归的正规化最低平方最小化。特别是,我们检查第二个方案的数字稳定性,并在Sinc内核的特殊情况下提供其趋同率。最后,我们用各种数字模拟来说明拟议的方法。