The main purpose of the paper is to study sharp estimates of approximation of periodic functions in the Hölder spaces H p for all 0 < p ≤ ∞ and 0 < α ≤ r. By using modifications of the classical moduli of smoothness, we give improvements of the direct and inverse theorems of approximation and prove the criteria for the precise order of decrease of the best approximation in these spaces. Moreove...

The paper presents upper estimates of the error of weighted and unweighted simultaneous approximation by the Bernstein operators and their iterated Boolean sums. The estimates are stated in terms of the Ditzian-Totik modulus of smoothness or appropriate K-functionals. AMS classification: 41A10, 41A17, 41A25, 41A28, 41A35, 41A36.

Interrelations between smoothness of a continuum at a point, pointwise smoothness, the property of Kelley at a point and local connectedness are studied in the paper.

The exterior sphere condition is compared to proximal smoothness, and examples are provided, which show that the two properties are not necessarily equivalent. Then conditions are given under which equivalence holds, and an open question involving the union of uniform closed balls property is stated in the form of a conjecture.

The purpose of this present paper is to give an equivalent theorem on Lupaş operators with ω φλ(f, t), where ω r φλ(f, t) is Ditzian-Totik modulus of smoothness for linear combination of Lupaş operators.

We consider reheating after inflation with a non-standard kinetic term. We show that the difference in the kinetic term in different Hubble patches inherited from the long-wavelength moduli inhomogeneities may generate a significant level of non-Gaussianity after inflation.

We present a characterization of the approximation errors of the Post-Widder and the Gamma operators in Lp[0, ∞), 1 ≤ p ≤ ∞, with a weight x γ for any real γ. Two types of characteristics are used – weighted K-functionals of the approximated function itself and the classical fixed step moduli of smoothness taken on a simple modification of it.

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also anal...

This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1 − 1/e ≈ .63, derived in Syrgkanis and Tardos (2013) using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the wor...