We show the sharpness of the modulus of continuity of a function f for which the domain lying above the graph of f satisfies the global boundary Harnack principle, with the aid of precise estimates of the Poisson integrals with respect to the Helmholtz equation in the half space.

Let ω(h, k) be a modulus of continuity, that is, ω(h, k) is a continuous function on the square [0, 2π] × [0, 2π], nondecreasing in each variable, and possessing the following properties: ω(0, 0) = 0, ω(t1 + t2, t3) ≤ ω(t1, t3) + ω(t2, t3), ω(t1, t2 + t3) ≤ ω(t1, t2) + ω(t1, t3). Yu ([3]) introduced the following classes of functions: HH := {f(x, y) : ‖f(x, y)− f(x+ h, y)− f(x, y + k) + f(x+ h,...

We study adaptive estimation of linear functionals over a collection of finitely many parameter spaces.A between class modulus of continuity, a geometric quantity, is introduced and is shown to be instrumental in characterizing the degree of adaptability over two parameter spaces in the same way that the usual modulus of continuity captures the minimax difficulty of estimation over a single par...

s at ICCAM 2012 Approximation by Chlodowsky type Jakimovski-Leviatan operators Ibrahim BÜYÜKYAZICI Ankara University Tandogan, Ankara Turkey [email protected] Joint work with: H. TANBERKAN, Ç.ATAKUT, S. KIRCI SERENBAY We introduce a generalization of the Jakimovski-Leviatan operators constructed by A.Jakimovski and D. Leviatan and the theorems on convergence and the degree of convergence a...

New modified Schurer-type q-Bernstein Kantorovich operators are introduced. The local theorem and statistical Korovkin-type approximation properties of these operators are investigated. Furthermore, the rate of approximation is examined in terms of the modulus of continuity and the elements of Lipschitz class functions.

In the space of continuous functions of a real variable, the set of nowhere dilferentiable functions has long been known to be topologically "generic". In this paper it is shown further that in a measure theoretic sense (which is different from Wiener measure), "almost every" continuous function is nowhere dilferentiable. Similar results concerning other types of regularity, such as Holder cont...

One of the important and basic topics in the theory of classical point set topology and several branches of mathematics, which have been researched by many authors, is continuity of functions. This concept has been extended to the setting of I-continuity of functions. Janković and Hamlett [1, 2] introduced the notion of I-open sets in topological spaces. Abd El-Monsef et al. [3] further investi...

We consider the class of integral operators Qφ on L (R+) of the form (Qφf)(x) = ∫ ∞ 0 φ(max{x, y})f(y)dy. We discuss necessary and sufficient conditions on φ to insure that Qφ is bounded, compact, or in the Schatten–von Neumann class Sp, 1 < p < ∞. We also give necessary and sufficient conditions for Qφ to be a finite rank operator. However, there is a kind of cut-off at p = 1, and for membersh...

Let T be an ergodic automorphism of the d-dimensional torus T, and f be a continuous function from T to R. On the probability space T equipped with the Lebesgue-Haar measure, we prove the weak convergence of the sequential empirical process of the sequence (f ◦T )i≥1 under some condition on the modulus of continuity of f . The proofs are based on new limit theorems and new inequalities for non-...

Compositional reasoning over probabilistic systems wrt. behavioral metric semantics requires the language operators to be uniformly continuous. We study which SOS specifications define uniformly continuous operators wrt. bisimulation metric semantics. We propose an expressive specification format that allows us to specify operators of any given modulus of continuity. Moreover, we provide a meth...