### A companion of Ostrowski's inequality for mappings whose first derivatives are bounded and applications in numerical integration

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Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. ${L}^{p}$, 1 ≤ p ≤ ∞) sense at $x\in {\mathbb{R}}^{m}$ if there are numbers ${f}_{\alpha}\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum}_{\left|\alpha \right|\le n}{f}_{\alpha}\left(x\right){h}^{\alpha}/\alpha !$ is $O\left({h}^{u}\right)$ in the approximate (resp. ${L}^{p}$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or ${L}^{p}$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and f = g on Π....

We consider complex-valued functions f ∈ L¹(ℝ), and prove sufficient conditions in terms of f to ensure that the Fourier transform f̂ belongs to one of the Lipschitz classes Lip(α) and lip(α) for some 0 < α ≤ 1, or to one of the Zygmund classes zyg(α) and zyg(α) for some 0 < α ≤ 2. These sufficient conditions are best possible in the sense that they are also necessary in the case of real-valued functions f for which either xf(x) ≥ 0 or f(x) ≥ 0 almost everywhere.

It is shown that every bi-Lipschitz bijection from Z to itself is at a bounded L1 distance from either the identity or the reflection.We then comment on the group-theoretic properties of the action of bi-Lipschitz bijections.

We study the bi-Lipschitz embedding problem for metric compacta hyperspaces. We observe that the compacta hyperspace K(X) of any separable, uniformly disconnected metric space X admits a bi-Lipschitz embedding in ℓ². If X is a countable compact metric space containing at most n nonisolated points, there is a Lipschitz embedding of K(X) in ${\mathbb{R}}^{n+1}$; in the presence of an additional convergence condition, this embedding may be chosen to be bi-Lipschitz. By way of contrast, the hyperspace K([0,1]) of the...

It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure $CFL{(}_{r})$ of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space ${}_{r}$ of diameter r, is (isometrically if r = +∞) isomorphic to the space $LIP{(}_{r})$ of equivalence classes of all real-valued Lipschitz maps on ${}_{r}$. The space of all signed (real-valued) Borel measures on ${}_{r}$ is isometrically embedded in the dual space of $CFL{(}_{r})$ and it is shown that the image of the embedding...

We give a characterization of the globally Lipschitzian composition operators acting in the space $B{V}_{p}^{2}[a,b]$

We show that the Hölder exponent and the chirp exponent of a function can be prescribed simultaneously on a set of full measure, if they are both lower limits of continuous functions. We also show that this result is optimal: In general, Hölder and chirp exponents cannot be prescribed outside a set of Hausdorff dimension less than one. The direct part of the proof consists in an explicit construction of a function determined by its orthonormal wavelet coefficients; the optimality is the direct consequence...

We study the relationship between the distortion function ${\Phi}_{K}$ and normalized quasisymmetric mappings. This is part of a new method for solving the boundary values problem for an arbitrary K-quasiconformal automorphism of a generalized disc on the extended complex plane.