Volume 26, Issue 2

It is proved that any function holomorphic in a real, non-degenerate Weil polyhedron G and continuous in its closure ¯ G can be uniformly approximated by functions holomorphic in a neighborhood of ¯ G . Besides, it is proved that such functions can be approximated by polynomials if G is a polynomial polyhedron.

We analyse certain dual pairs of orthogonal systems of rational matrix-valued functions with poles not located on the unit circle. The rational matrix-valued functions belonging to such kind of pairs are recursively connected similar as the functions of the first and the second kind in the classical theory of orthogonal polynomials on the unit circle. We present some basic facts on these dual pairs and, in particular, we show that the rational matrix-valued functions in question admit specific integral representations.

Suppose that u ( z ) and v ( z ) are subharmonic functions in the plane, of order at most ρ, where 0 < ρ < 1/2, and that u ( z ) is nonconstant. Let φ( z ) = u ( z ) − v ( z ), and suppose that the deficiency δ of φ( z ) satisfies 0 ≤ 1 − δ < cos πρ. It is shown that, given σ such that both ρ < σ < 1/2 and 0 ≤ 1 − δ < cos πσ, we have A ( r ,φ)/ B ( r ,φ) > (cos πσ − (1 − δ))/(1 − (1 − δ)cos πσ), for all r in a set of lower logarithmic density at least 1 − ρ/σ.

We consider critical immersions of singular variational integrals of the type E α ( X ) = ∫ M X α n + 1 dA , α > 0. We construct quadratic forms which are subharmonic on any stationary hypersurface using methods of Dierkes [4] and Hildebrandt [9]. This leads to enclosure theorems which generalize well-known results of Böhme, Hildebrandt, and Tausch [2] for minimal surfaces under gravitational forces. Finally, we discuss the application of these estimates to the corresponding curvature flow problem.

Recently, some new quasi-interpolants have been introduced to accelerate the convergence of the classical approximation operators. In this paper we consider Bernstein–Kantorovich quasi-interpolants K n (2 r − 1) ( f,x ) and obtain direct, inverse and equivalence theorems in terms of the Ditzian's moduli of smoothness .

We consider a boundary value problem in the unit disk for an elliptic equation of order N with constant coefficients. We suppose, that the characteristic equation has exactly M pairs of complex conjugate roots (0 ≤ 2 M ≤ N ). The solution of this problem belongs to the class of N times continuously differentiable functions, which with up to the ( N − M − 1)-th order derivatives satisfy the Hölder condition in the closed disk. We prove, that the inhomogeneous problem has a solution for arbitrary boundary functions, and the corresponding homogeneous problem (when all boundary functions are identically zero) has ( N — 2 M ) 2 linearly independent (over the field of real numbers) solutions. The solutions of the homogeneous problem are the purely imaginary polynomials of order 2( N − M ) − 2.