Volume 23, Issue 1

In our companion paper [3] we studied a number of different Sobolev spaces on a general (non-Lipschitz) open subset Ω of ℝn{\mathbb{R}^{n}}, defined as closed subspaces of the classical Bessel potential spaces Hs⁢(ℝn){H^{s}(\mathbb{R}^{n})} for s∈ℝ{s\in\mathbb{R}}. These spaces are mapped by the restriction operator to certain spaces of distributions on Ω. In this note we make some observations about the relation between these spaces of global and local distributions. In particular, we study conditions under which the restriction operator is or is not injective, surjective and isometric between given pairs of spaces. We also provide an explicit formula for minimal norm extension (an inverse of the restriction operator in appropriate spaces) in a special case.

In the present paper, the coefficients characterizations of generalized type Tm⁢(f;α,α){T_{m}(f;\alpha,\alpha)} of entire transcendental functions f of several complex variables m (m≥2){m\geq 2)} for slow growth have been obtained in terms of the sequence of best polynomial approximations of f in the Hardy Banach spaces Hq⁢(Um){H_{q}(U^{m})} and in the Banach spaces Bm⁢(p,q,λ){B_{m}(p,q,\lambda)}. The presented work is the extension and refinement of the corresponding assertions made by Vakarchuk and Zhir [22, 20, 21, 23, 24, 25], Gol’dberg [4] and Sheremeta [17, 18] to the multidimensional case.

We give a new proof and discuss an extension of Jack’s lemma for polynomials.

In the paper we discuss the functional Φf⁢(μ)≡a2⁢a4-μ⁢a32{\Phi_{f}(\mu)\equiv a_{2}a_{4}-\mu a_{3}^{2}} for functions in the class ℛ⁢(α){\mathcal{R}(\alpha)}, α∈[0,1){\alpha\in[0,1)}. This class consists of analytic functions which satisfy the condition Re⁡f′⁢(z)>α{\operatorname{Re}f^{\prime}(z)>\alpha} for all z in the unit disk Δ. We show that the conjecture of Hayami and Owa [1], that is, |Φf⁢(μ)|≤(1-α)2⋅max⁡{12-49⁢μ,49⁢μ}{|\Phi_{f}(\mu)|\leq(1-\alpha)^{2}\cdot\max\{\tfrac{1}{2}-\tfrac{4}{9}\mu,% \tfrac{4}{9}\mu\}} for all f∈ℛ⁢(α){f\in\mathcal{R}(\alpha)} and μ∈ℝ{\mu\in\mathbb{R}}, is false. Moreover, we find estimates of |Φf⁢(μ)|{|\Phi_{f}(\mu)|} that improve the results obtained by Hayami and Owa.

We investigate classes of k -uniformly convex functions which generalize the class UCV{\mathrm{UCV}} introduced by Goodman in 1991. We solve the problem of finding the largest α≥0{\alpha\geq 0} such that the class of k -uniformly convex functions is contained in the class of starlike functions of order α.

In a von Neumann factor ℳ{\mathcal{M}} of type II∞{{}_{\infty}} with semifinite faithful normal trace τ, the q -th Schatten-type norm ∥⋅∥q{\|\cdot\|_{q}} of an operator A∈ℳ{A\in\mathcal{M}} is defined as ∥A∥q=(τ⁢(|A|q))1q{\|A\|_{q}=(\tau(|A|^{q}))^{\frac{1}{q}}}. We will prove that for every self-adjoint operator H∈ℳ{{H}\in\mathcal{M}} and any numbers q>1{q>1}, ε>0{\varepsilon>0}, there exists a self-adjoint perturbation operator A∈ℳ{A\in\mathcal{M}} such that ∥A∥<ε{\|A\|<\varepsilon}, ∥A∥q<ε{\|A\|_{q}<\varepsilon}, and H+A{H+A} has a pure point spectrum. We will also discuss the possibility of transferring this result onto other von Neumann factors.

We deal with the wave equation with assigned moving boundary (0<x<a⁢(t){0<x<a(t)}) upon which Dirichlet or mixed boundary conditions are specified. Here a⁢(t){a(t)} is assumed to move slower than light and periodically. Moreover, a is continuous, piecewise linear with two independent parameters. Our major concern will be an observation problem which is based measuring, at each t>0{t>0}, of the transverse velocity at a⁢(t){a(t)}. The key to the results is the use of a reduction theorem by Yoccoz [14].

In this article, we study the best approximation in quotient probabilistic normed space. We define the notion of quotient space of a probabilistic normed space, then prove some theorems of approximation in quotient space are extended to quotient probabilistic normed space.