On a consistency of orthogonal series estimators with respect to Jacobi polynomials system

Consider a nonparametric regression model $ Y_{i} =m(X_{i} )+\varepsilon _{i} \, ,\, i=1,\dots,n, $ where $m(x)$ is the unknown regression function to be estimated, $\left\{\left(X_{i} ,Y_{i} \right)\right\}_{i=1}^{n} $ is a dataset and $\{ \varepsilon _{i} \} _{i=1}^{n} $ are observation errors. Suppose that the regression function can be represented as a Fourier series $ m\left(x\right)=\sum _{j=0}^{\infty }\beta _{j} \varphi _{j} \left(x\right), $ where the system of functions $\left\{\, \varphi _{j} \left(x\right)\right\}_{j=0}^{\, \infty } $ constitutes an orthonormal basis on $[-1,1]$, with respect to inner product $ \left(f ,g \right)=\int _{-1}^{1}f\left(x\right)g\left(x\right)\rho(x)\, dx, $ and $\{\beta _{j}\}$ are Fourier coeffcients. Next assume that observations $\{ Y _{i} \} _{i=1}^{n} $ have been taken at equidistant points $\{ X _{i} \} _{i=1}^{n} $ over the interval $[-1, 1]$ and let $\left\{A_{i} \right\}_{i=1}^{n} $ be a set of disjoint intervals such that $\cup_{i=1}^n A_{i}=[-1,1]$ and $X_{i} \in A_{i} $, $i=1,...,n$. Put $ \hat{m}_{N(n)} \left(x\right)=\sum _{j=0}^{N\left(n\right)}\hat{\beta }_{j} \varphi _{j} \left(x\right), \;\hat{\beta }_{j} =\sum _{i=1}^{n}Y_{i} \int _{A_{i} }\varphi _{j} \left(x\right)\rho(x)\,dx, $ where $N(n)$ is a suitable finite number. This estimator is called an orthogonal series estimator of $m(x)$.
In the present paper, we give the consistency condition for $\hat{m}(x)$ provided that the regression function $m(x)$ is Lipschitz continuous and $ \varphi _{j} \left(x\right)= P_{j}^{\, (\alpha ,\beta )} (x),\,j=0,1,\dots, $ is the Jacobi orthonormal polynomials system with certain restrictions for exponents $\alpha,\, \beta $. The main result is as follows.
Theorem 1. Suppose that the following conditions are satisfied:
i) $\mathsf{E}\varepsilon _{i} =0$, $\mathsf{E}(\varepsilon _{i}\varepsilon _{j}) =0$, $i\neq j$, and $\mathsf{E}\varepsilon _{i}^2 <C$, $i=1,\dots,n$;
ii) $ m(\cdot) \in \mathrm{Lip}_M1; $
iii) $ p:=\min\{\alpha;\beta\}\ge-1/2; $
iv) $ (N(n))^2=o\left\{A_n(\alpha;\beta)\right\},\;n\to\infty, $ where $A_n(\alpha,\beta)=n$, if $p>-1/2$ and $A_n(\alpha,\beta)=n/\log n$, if $p=-1/2$. Then $\hat{m}_{N} \left(x\right)\stackrel{p}{\longrightarrow} m\left(x\right), \;N\left(n\right)\to \infty$ for every $x\in(-1,1)$.
Theorem 2. Let the conditions {i)–iii)} of previous theorem are satisfied, $q=\max\{\alpha;\beta\}<1/2$, and $ \left(N(n)\right)^{2q+3}=o\left\{A_n(\alpha,\beta)\right\},\;n\to\infty. $ Then $\hat{m}_{N} \left(x\right)\stackrel{p}{\longrightarrow} m\left(x\right)$, $N\left(n\right)\to \infty$, for every $x\in[-1,1]$.