  In the first part of this work, we develop a novel scheme for solving
nonparametric regression problems. That is the approximation of possibly low
regular and noised functions from the knowledge of their approximate values
given at some random points. Our proposed scheme is based on the use of the
pseudo-inverse of a random projection matrix, combined with some specific
properties of the Jacobi polynomials system, as well as some properties of
positive definite random matrices. This scheme has the advantages to be stable,
robust, accurate and fairly fast in terms of execution time. In particular, we
provide an $L_2$ as well as an $L_2-$risk errors of our proposed nonparametric
regression estimator. Moreover and unlike most of the existing nonparametric
regression estimators, no extra regularization step is required by our proposed
estimator. Although, this estimator is initially designed to work with random
sampling set of uni-variate i.i.d. random variables following a Beta
distribution, we show that it is still works for a wide range of sampling
distribution laws. Moreover, we briefly describe how our estimator can be
adapted in order to handle the multivariate case of random sampling sets. In
the second part of this work, we extend the random pseudo-inverse scheme
technique to build a stable and accurate estimator for solving linear
functional regression (LFR) problems. A dyadic decomposition approach is used
to construct this last stable estimator for the LFR problem. Alaso, we give an
$L_2-$risk error of our proposed LFR estimator. Finally, the performance of the
two proposed estimators are illustrated by various numerical simulations. In
particular, a real dataset is used to illustrate the performance of our
nonparametric regression estimator.
