Vol 217, No 1 (2016)

Let σ > 0, m, r ∈ ℕ, m ≥ r, let S_σ,m be the space of splines of order m and minimal defect with nodes \( \frac{j\pi }{\sigma } \) (j ∈ ℤ), and let A_σ,m(f)_p be the best approximation of a function f by the set S_σ,m in the space L_p(ℝ). It is known that for p = 1,+∞,

where K_r are the Favard constants. In this paper, linear operators X_σ,r,m with values in S_σ,m such that for all p ∈ [1,+∞] and f ∈ W_p^(r)(ℝ), are constructed. This proves that the upper bounds indicated above can be achieved by linear methods of approximation, which was previously unknown. Bibliography: 21 titles.

Some distortion theorems for circumferentially mean p-valent functions are proved using the symmetrization method. The cases of functions with a zero of order p at the origin, functions having no zeros, and functions with Montel’s normalization are considered. The equality cases in the estimates obtained are described. Bibliography: 10 titles.

The paper considers strong approximation of continuous functions by positive operators. Estimates in terms of the modulus of continuity and its convex majorant are established. Bibliography: 4 titles.

With the use of differentiation of exchanged toric developments, an infinite sequence of twodimensional toric tilings is constructed. The karyon of these tilings is proved to be a bounded remainder set.

The logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been established by Finner and Roters in 1997 as a corollary of a rather difficult result, based on generalized reproductive property of certain distributions. In the first part of this paper, a direct analytic proof of the logarithmic concavity/convexity mentioned above is presented. These results are strengthened in the second part, where it is proved that the power series coefficients of the generalized Turán determinants formed by the parameter shifts of the normalized incomplete beta function have constant sign under some additional restrictions. The method of proof suggested also leads to various other new facts, which may be of independent interest. In particular, linearization formulas and two-sided bounds for the above-mentioned Turán determinants, and also two identities of combinatorial type, which we believe to be new, are established.

An extension of the method of modules of curve families to extremal decomposition problems in which the associated quadratic differentials have poles of arbitrary orders is considered.