In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the type \begin{equation} \alpha=\displaystyle\frac{A\alpha^{q}+B}{C\alpha^{q}} \end{equation} where $(A, B, C)\in (\mathbb{F}_{q}[X])^{2}\times\mathbb{F}_{q}^{\star}[X]$. In particular, we will give, under some conditions on the polynomials $A$, $B$ and $C$, well approximated elements satisfying this equation.

We show that the following $K_0$-monoid properties of $C^*$-algebras in the class $\Omega$ are inherited by simple unital $C^*$-algebras in the class $TA\Omega$: (1) weak comparability, (2) strictly unperforated, (3) strictly cancellative.

We prove the existence of an approximation function for holomorphic solutions of a system of real analytic equations. For this we use ultraproducts and Weierstrass systems introduced by J. Denef and L. Lipshitz. We also prove a version of the Płoski smoothing theorem in this case.

We show that any Lipschitz projection-valued function $p$ on a connected closed Riemannian manifold can be approximated uniformly by smooth projection-valued functions $q$ with Lipschitz constant close to that of $p$. This answers a question of Rieffel.

We consider approximation of multivariate functions in Sobolev spaces by high order Parzen windows in a non-uniform sampling setting. Sampling points are neither i.i.d. nor regular, but are noised from regular grids by non-uniform shifts of a probability density function. Sample function values at sampling points are drawn according to probability measures with expected values being values of the approximated function. The approximation orders are estimated by means of regularity of the approximated function, the density function, and the order of the Parzen windows, under suitable choices of the scaling parameter.

For a given convex body $K$ in ${\mathbb R}^d$, a random polytope $K^{(n)}$ is defined (essentially) as the intersection of $n$ independent closed halfspaces containing $K$ and having an isotropic and (in a specified sense) uniform distribution. We prove upper and lower bounds of optimal orders for the difference of the mean widths of $K^{(n)}$ and $K$ as $n$ tends to infinity. For a simplicial polytope $P$, a precise asymptotic formula for the difference of the mean widths of $P^{(n)}$ and $P$ is obtained.

Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.

New necessary and sufficient conditions are established for Banach spaces to have the approximation property; these conditions are easier to check than the known ones. A shorter proof of a result of Grothendieck is presented, and some properties of a weak version of the approximation property are addressed.

A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed.

In the present paper, we introduce a modification of the Meyer-K\"{o}nig and Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and $f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r-$th order generalization of our operators and study their approximation properties.

Continuous mappings defined from compact subsets $K$ of complex Euclidean space $\cc^n$ into complex projective space $\pp^m$ are approximated by rational mappings. The fundamental tool employed is homotopy theory.

On a locally rectifiable arc going to infinity, each continuous function can be approximated by entire functions.