We study properties of generalized $K$-functionals and generalized moduli of smoothness in $L_p(\R)$ spaces with $1 \le p \le +\infty$. We obtain the direct Jackson type estimate and the inverse Bernstein type estimate for them. We state equivalence between approximation error of convolution integrals generated by an arbitrary generator with compact support generalized $K$-functionals generated by homogeneous function and generalized moduli of smoothness generated by $2\pi$-periodic generator subject to equivalence of their generators. We show that generalized $K$-functionals and generalized moduli of smoothness contain, as their special cases many well-known constructions of $K$-functionals and moduli of smoothness with an appropriate choice of the generators.
In the first part of the paper, it is proved that for 1 < p < ∞ the couple (K θ p , K θ ∞ ) of coinvariant subspaces of the shift operator on the unit circle is K-closed in the couple (L p (T),L∞ (T)). This property underlies basically all problems of real interpolation for the first couple. Also, a weighted analog of the above statement is established.
In the second part it is shown that, given two closed ideals I and J in a uniform algebra such that the complex conjugate of I ∩ J is not included in some of them, the sum I + J̅ is not closed.
Though the methods of study in the two parts are quite different, the topics are related by the fact that the question treated in the second part emerged during the work on the first.
New results and a survey of known results are presented on various concepts of negligible sets in infinite-dimensional spaces.
According to a classical result of E. Calabi any hyperbolic affine hypersphere endowed with its natural Hessian metric has a non-positive Ricci tensor. The affine hyperspheres can be described as the level sets of solutions to the “hyperbolic” toric Kähler–Einstein equation eΦ = detD2Φ on proper convex cones. We prove a generalization of this theorem by showing that for every Φ solving this equation on a proper convex domain Ω the corresponding metric measure space (D2Φ, eΦdx) has a non-positive Bakry–Émery tensor. Modifying the Calabi computations we obtain this result by applying the tensorial maximum principle to the weighted Laplacian of the Bakry–Émery tensor. Our computations are carried out in a generalized framework adapted to the optimal transportation problem for arbitrary target and source measures. For the optimal transportation of the log-concave probability measures we prove a third-order uniform dimension-free apriori estimate in the spirit of the second-order Caffarelli contraction theorem, which has numerous applications in probability theory.
We show that if ϕ: R → R is a continuous mapping and the set of nonlinearity of ϕ has nonzero Lebesgue measure, then ϕ maps bijectively a certain set that contains arbitrarily long arithmetic progressions onto a certain set with distinct sums of pairs.
We establish several Jackson type inequalities with explicit constants for spline approximation of functions deﬁned on the real ax is. The inequalities for the ﬁrst modulus of continuity of odd derivatives are sharp. We also obtain inequalities for high-order moduli of continuity of a function itself. One of the inequalities for th e second modulus of continuity is sharp. Up to the present paper no estimates for spline approximation on the axis in terms of high-order moduli of continuity, with constants written explicitly, were known.