In the present paper, we study the polynomial approximation of analytic functions of several complex variables. The characterizations of generalized type of analytic functions of several complex variables have been obtained in terms of approximation and interpolation errors.

We shall generalize the results of [9] about characterization of isomorphisms on quasi-Banach algebras by providing integral type conditions. Also, we shall give some new results in this way and finally, give a result about hybrid fixed point of two homomorphisms on quasi-Banach algebras.

In this paper, we shall introduce and characterize simultaneous quasi-Chebyshev (and weakly-Chebyshev) subspaces of normed spaces with respect to a bounded set $S$ by using elements of the dual space.

The Lipschitz classes $Lip(\alpha,M)$, $0 < \alpha \leq1$ are defined for Orlicz space generated by the Young function $M$, and the degree of approximation by matrix transforms of $f \in Lip(\alpha,M)$ is estimated by $n^{−\alpha}$.

The bivariate interpolation in two dimensional space $\mathbf{R}^2$ is more complicated than that in one dimensional space $\mathbf{R}$, because there is no Haar space of continuous functions in $\mathbf{R}^2$. Therefore, the bivariate interpolation has not a unique solution for a set of arbitrary distinct pairwise points. In this work, we suggest a type of basis which depends on the points such that the bivariate interpolation has the unique solution for any set of distinct pairwise points. In this case, the matrix of bivariate interpolation has the semi inherited factorization.

If $p(z)$ is a polynomial of degree $n$ having all its zeros on $|z| = k$, $k \leq 1$, then it is proved^[5] that $$\max_{|z|=1}|p′(z)| \leq\frac{n}{k^{n−1}+k^n}\max_{|z|=1}|p(z)|.$$In this paper, we generalize the above inequality by extending it to the polar derivative of a polynomial of the type $p(z) = c_nz^n +\sum\limits_{j=\mu}^{n}c_{n-j}z^{n-j}$, $1 \leq \mu  \leq n$. We also obtain certain new inequalities concerning the maximum modulus of a polynomial with restricted zeros.

For maximal multilinear Calderón-Zygmund singular integral operators, the sharp maximal function estimate and some weighted norm inequalities are obtained.

In the present paper we study properties of Szász-Durrmeyer operators. These operators are introduced in [5] and generalize the integral operators proposed by S. M. Mazhar and V. Totik in [12]. We also generalize some results obtained by M. Heilmann^[6] and D.-X. Zhou^[16].

We will show bounds for commutators of multilinear fractional integral operators with some homogeneous kernels.

For a real valued function $f$ defined on a finite interval $I$ we consider the problem of approximating $f$ from null spaces of differential operators of the form $L_n(\psi) =\sum\limits_{k=0}^{n}a_k\psi^{(k)}$, where the constant coefficients $a_k \in R$ may be adapted to $f$.
We prove that for each $f \in C^{(n)}(I)$, there is a selection of coefficients $\{a_1, \cdots,a_n\}$ and a corresponding linear combination$$S_n( f , t) =\sum_{k=1}^nb_k e^{\lambda_{k^t}}$$of functions $\psi_k(t) = e^{\lambda_kt}$ in the nullity of $L$ which satisfies the following Jackson’s type inequality: $$\|f^{(m)}-S_n^{(m)}(f,t)\|_{\infty}\le \frac{|I|^{1/q}e^{|\lambda_n||I|}}{|a_n|2^{n-m-1/p}|\lambda_n|^{n-m-1}}\|L_n(f)\|_p$$ where $|\lambda_n| = \max\limits_k |\lambda_k|$, $0 \leq m \leq n−1,$ $p,q \geq 1$, and $\frac{1}{p}+\frac{1}{q}= 1.$
For the particular operator $M_n( f ) = f +1/(2n)! f ^{(2n)}$ the rate of approximation by the eigenvalues of $M_n$ for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.