We present a self-contained and modern survey of some existing quasi-sure results via the connection to the Brownian sheet. Among other things, we prove that quasi-every continuous function: (i) satisfies the local law of the iterated logarithm; (ii) has Lévy’s modulus of continuity for Brownian motion; (iii) is nowhere differentiable; and (iv) has a nontrivial quadratic variation. We also pres...

Given a diffeomorphism of the interval, consider the uniform norm of the derivative of its n-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in geometry of the diffeomorphisms groups. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the mod...

Abstract. We study inequalities between general integral moduli of continuity of a function and the tail integral of its Fourier transform. We obtain, in particular, a refinement of a result due to D. B. H. Cline [2] (Theorem 1.1 below). We note that our approach does not use a regularly varying comparison function as in [2]. A corollary of Theorem 1.1 deals with the equivalence of the two-side...

In this paper two kinds of Kantorovich-type q-Bernstein-Stancu operators are introduced, and the statistical approximation properties of these operators are investigated. Furthermore, by means of modulus of continuity, the rates of statistical convergence of these operators are also studied. MSC: 41A10; 41A25; 41A36

A uniqueness theorem is proven for the problem of the recovery of a complex valued compactly supported 2-D function from the modulus of its Fourier transform. An application to the phase problem in optics is discussed.

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.