Vol 243, No 6 (2019)
- Year: 2019
- Articles: 20
- URL: https://journals.rcsi.science/1072-3374/issue/view/15041
Article
Resolvent Kernels of Self-Adjoint Extensions of the Laplace Operator on the Subspace of Solenoidal Vector Functions
Abstract
The Laplace operator on the subspace of solenoidal vector functions of three variables vanishing at the origin together with first derivatives is a symmetric operator with deficiency indices (3). Krein’s theory allows one to derive an expression for the resolvent kernel of a self-adjoint extension of the operator in question as a sum of the Green’s function of the vector Laplace operator and some additional kernel of finite rank.
On the Absolute Convergence of Fourier–Haar Series in the Metric of Lp(0, 1), 0 < p < 1
Abstract
It is proved that for every ∈ > 0 there exists a measurable set E ⊂ [0, 1] with measure |E| > 1 − ∈ such that for every function f(x) ∈ L[0, 1] one can find a function g(x) ∈ L[0, 1] coinciding with f(x) on E such that its Fourier–Haar series absolutely converges in the metric of Lp(0, 1), 0 < p < 1.
On Products of Weierstrass Sigma Functions
Abstract
We prove the following result. Let f : ℂ → ℂ be an even entire function. Assume that there exist ????j, βj : ℂ → ℂ with
Then f(z) = σL(z) · σΛ(z) · eAz2+C where L and Λ are lattices in ℂ, σL is the Weierstrass sigma function associated with the lattice L, and A,C ∈ ℂ.
Kernels of Toeplitz Operators and Rational Interpolation
Abstract
The kernel of a Toeplitz operator on the Hardy class H2 in the unit disk is a nearly invariantsubspace of the backward shift operator, and, by D. Hitt’s result, it has the form g · Kω where ω is an inner function, Kω = H2 ⊝ ωH2, and g is an isometric multiplier on Kω. We describe the functions ω and g for the kernel of the Toeplitz operator with symbol .\( \overline{\theta}\varDelta \) where θ is an inner function and Δ is a finite Blaschke product.
A Remark on Indicator Functions with Gaps in the Spectrum
Abstract
Developing a recent result of F. Nazarov and A. Olevskii, we show that for every subset a in ℝ of finite measure and every ε > 0 there exists b ⊂ ℝ with |b| = |a| and |(b \ a) ∪ (a \ b)|≤ε such that the spectrum of χb is fairly thin. A generalization to locally compact Abelian groups is also provided.
Correction Up to a Function with Sparse Spectrum and Uniformly Convergent Fourier Integral for the Group ℝn
Abstract
This is an ℝn-counterpart of certain considerations on a similar subject for compact Abelian groups exposed by P. Ivanishvili and the author in 2010. The main difference with that paper is that certain notions and results of measure theory should be invoked in the case of ℝn.
The Bellman Function for a Parametric Family of Extremal Problems in BMO
Abstract
Let I be an interval of the real line and 〈⋅〉I be the corresponding integral average. We describe the behavior of the Bellman function for the functional F(φ) = 〈f ∘ φ〉I, φ ∈ BMO(I), as f ranges over some parametric family of functions. Thereby, we once again demonstrate the power of the methods developed recently by V. I. Vasyunin, P. B. Zatitskiy, P. Ivanishvili, D. M. Stolyarov, and the author.
The Hausdorff Measure on n-Dimensional Manifolds in ℝm and n-Dimensional Variations
Abstract
We extend the notion of the variation Vf([a; b]) of a function f : [a; b] → ℝ to the variation Vf(A) of a continuous map f : G → ℝn, where G is an open subset of ℝn, over a set A ⊂ G of the form A = ∪i ∈ IKi where I is countable and all Ki are compact.
Let f : G → ℝm where G ⊂ ℝn with n ≤ m, and let f1, . . . , fm be the coordinate functions of f. For α = {i1, . . . , in} where 1 ≤ i1 < i2 < ⋯ < in ≤ m, let fα be the map with coordinate functions \( {f}_{i_1},\dots, {f}_{i_n} \). The main result of the paper states that if f is a continuous injective map, G is an open subset of ℝn, and a subset A ⊂ G has the form A = ∪i ∈ IKi where I is countable and all Ki are compact, then \( {V}_{f_{\alpha }}(A)\le {H}_n\left(f(A)\right) \) where \( {V}_{f_{\alpha }}(A) \) is the variation of fα over A and Hn is the n-dimensional Hausdorff measure in ℝm.
Stieltjes Integrals in the Theory of Harmonic Functions
Abstract
We study various Stieltjes integrals (Poisson–Stieltjes, conjugate Poisson–Stieltjes, Schwartz– Stieltjes, and Cauchy–Stieltjes integrals) and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert–Stieltjes integral. These results are valid for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands from the class CBV (countably bounded variation).
On the Boundary Behavior of Some Classes of Mappings
Abstract
We study the boundary behavior of closed open discrete mappings from the Sobolev and Orlicz–Sobolev classes in ℝn, n ≥ 3. It is proved that such a mapping f can be extended by continuity to a boundary point x0 ∈ ∂ D of a domain D ⊂ ℝn whenever its inner dilatation of order α > n− 1 has a majorant from the finite mean oscillation class at the point in question. Another sufficient condition for the existence of a continuous extension is the divergence of some integral. We also prove some results on the continuous extension of such a mapping to an isolated boundary point.
Stability of Nearly Optimal Decompositions in Fourier Analysis
Abstract
We consider the existence problem for near-minimizers for the distance functional (or E-functional in the interpolation terminology) that are stable under the action of certain operators. In particular, stable near-minimizers for the couple (L1, Lp) are shown to exist when the operator is the projection to wavelets and these wavelets satisfy only some weak decay conditions at infinity.
Interpolation Through Approximation in a Bernstein Space
Abstract
Let Bσ be the Bernstein space of entire functions of exponential type at most σ bounded on the real axis. Consider a sequence Λ = {zn}n∈ℤ, zn = xn + iyn, such that xn+1 − xn ≥ l > 0 and |yn| ≤ L, n ∈ ℤ. Using approximation by functions from Bσ, we prove that for any bounded sequence A = {an}n∈ℤ, |an| ≤ M, n ∈ ℤ, there exists a function f ∈ Bσ with σ ≤ σ0(l,L) such that f|Λ = A.
A Note on Approximation by Trigonometric Polynomials
Abstract
Let \( E=\underset{k=1}{\overset{n}{\cup }}\left[{a}_k,{b}_k\right]\subset \mathbb{R} \); if n > 1, then we assume that the segments [ak, bk] are pairwise disjoint. Assume that the following property holds: E ∩ (E + 2πν) = ∅, ν ∈ ℤ, ν ≠ 0. Denote by Hω + r(E) the space of functions f defined on E such that |f(r)(x2) − f(r)(x1)| ≤ cfω(|x2 − x1|), x1, x2 ∈ E, f(0) ≡ f. Assume that a modulus of continuity ω satisfies the condition
We find a constructive description of the space Hω + r(E) in terms of the rate of nonuniform approximation of a function f ∈ Hω + r(E) by trigonometric polynomials if E and ω satisfy the above conditions.
On the Sharpness of the Estimate in a Theorem Concerning the Half Smoothness of a Function Holomorphic in a Ball
Abstract
Let ????n be the unit ball and Sn be the unit sphere in ℂn, n ≥ 2. Let 0 < α < 1, and define a function f on as follows:
The main result of the paper is the following theorem: the function ζ ↦ |f(ζ)| on the unit sphere Sn belongs to the Hölder class Hα(Sn), while the function f does not belong to the Hölder class for any ε > 0.