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 $n$.

In this paper we introduce the notation of t-best approximatively compact sets, t-best approximation points, t-proximinal sets, t-boundedly compact sets and t-best proximity pair in fuzzy metric spaces. The results derived in this paper are more general than the corresponding results of metric spaces, fuzzy metric spaces, fuzzy normed spaces and probabilistic metric spaces.