Abstract:
In this paper we investigate weighted polynomial approximations with several
variables. Our study relates to the approximation for by weighted
polynomial. Then we will give some results relating to the Lagrange interpolation,
the best approximation, the Markov-Bernstein inequality and the Nikolskii-
type inequality.

Abstract:
In this paper, we investigate the weighted polynomial approximations with
several variables. Our study relates to the approximation for by weighted polynomials. Then we will estimate the degree of approximation.

Abstract:
Let , , where is an even function. In 2008 we have a relation of the orthonormal polynomial with respect to the weight ; , where and are some integrating functions for orthonormal polynomials . In this paper, we get estimates of the higher derivatives of and , which are important for estimates of the higher derivatives of .

Abstract:
Let , let be a continuous, nonnegative, and increasing function, and let be the orthonormal polynomials with the weight . For the zeros of we estimate , where is a positive integer. Moreover, we investigate the various weighted -norms ( ) of .

Abstract:
Lupas-type operators and Szász-Mirakyan-type operators are the modifications of Bernstein polynomials to infinite intervals. In this paper, we investigate the convergence of Lupas-type operators and Szász-Mirakyan-type operators on [0,∞).

Abstract:
Let ()∶=(1？2)？1/2 and ,() be the ultraspherical polynomials with respect to (). Then, we denote the Stieltjes polynomials with respect to () by ,

Abstract:
Let wρ(x):=|x|ρexp( Q(x)), ρ> 1/2, where Q∈C2:( ∞,∞)→[0,∞) is an even function. In 2008 we have a relation of the orthonormal polynomial pn(wρ2;x) with respect to the weight wρ2(x); pn′(x)=An(x)pn 1(x) Bn(x)pn(x) 2ρnpn(x)/x, where An(x) and Bn(x) are some integrating functions for orthonormal polynomials pn(wρ2;x). In this paper, we get estimates of the higher derivatives of An(x) and Bn(x), which are important for estimates of the higher derivatives of pn(wρ2;x).

Abstract:
Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow[0,\infty)$ be an even function. We consider the exponential weights $w(x)=e^{-Q(x)}$, $x\in \mathbb{R}$. In this paper we obtain a pointwise convergence theorem for the Fourier-type series with respect to the orthonormal polynomials $\left\{p_n(w^2;x)\right\}$.

Abstract:
Let $\mathbb{R}=(-\infty,\infty)$, and let $Q\in C^1(\mathbb{R}): \mathbb{R}\rightarrow \mathbb{R^+}=[0,\infty)$ be an even function, which is an exponent. We consider the weight $w_\rho(x)=|x|^{\rho} e^{-Q(x)}$, $\rho\geqslant 0$, $x\in \mathbb{R}$, and then we can construct the orthonormal polynomials $p_{n}(w_\rho ^2;x)$ of degree n for $w_\rho ^2(x)$. In this paper we obtain $L_p$-convergence theorems of even order Hermite-Fej\'er interpolation polynomials at the zeros $\left\{x_{k,n,\rho}\right\}_{k=1}^n$ of $p_{n}(w_\rho ^2;x)$.