We show that in the space C[-1,1] there exists an orthogonal algebraic polynomial basis with optimal growth of degrees of the polynomials.

Chebyshevian box splines were introduced in [5]. The purpose of this paper is to show some new properties of them in the case when the weight functions $w_j$ are of the form $w_j\left(x\right)=W_j\left(v_{n+j}·x\right)$, where the functions $W_j$ are periodic functions of one variable. Then we consider the problem of approximation of continuous functions by Chebyshevian box splines.

We construct a $C^k$ piecewise differentiable function that is not $C^k$ piecewise analytic and satisfies a Jackson type estimate for approximation by Lagrange interpolating polynomials associated with the extremal points of the Chebyshev polynomials.

A characterization of a generalized order of analytic functions of several complex variables by means of polynomial approximation and interpolation is established.

In this paper we extend the Duman-King idea of approximation of functions by positive linear operators preserving $e_k\left(x\right)=x^k$, $k=0,2$. Using a modification of certain operators $L_n$ preserving $e_0$ and $e_1$, we introduce operators $L_n^{*}$ which preserve $e_0$ and $e_2$ and next we define operators $L_{n;r}^{*}$ for $r$-times differentiable functions. We show that $L_n^{*}$ and $L_{n;r}^{*}$ have better approximation properties than $L_n$ and $L_{n;r}$.

Let ${𝕋}_n$ be the space of all trigonometric polynomials of degree not greater than $n$ with complex coefficients. Arestov extended the result of Bernstein and others and proved that $\parallel (1/n)T_n^{\text{'}}{\parallel }_p\le {\parallel T_n\parallel }_p$ for $0\le p\le \infty$ and $T_n\in {𝕋}_n$. We derive the multivariate version of the result of Golitschek and Lorentz $${∥{\left|T_{n}cos\alpha +\frac{1}{n}\nabla T_{n}sin\alpha \right|}_{l_{\infty }^{\left(m\right)}}∥}_p\le {\parallel T_n\parallel }_p,0\le p\le \infty$$ for all trigonometric polynomials (with complex coeffcients) in $m$ variables of degree at most $n$.