Том 234, № 3 (2018)

In this paper, we construct anticliques for noncommutative operator graphs generated by generalized Pauli matrices. It is shown that application of entangled states for the construction of the code space K allows one to substantially increase the dimension of a noncommutative operator graph for which the projection on K is an anticlique.

We obtain certain sufficient conditions under which the couple of weighted Hardy spaces

\( \left({H}_r\left({w}_1\left(\cdot, \cdot \right)\right),{H}_s\left({w}_2\left(\cdot, \cdot \right)\right)\right) \)

on the two-dimensional torus ????² is K-closed in the couple (L_r(w₁( · , · )), L_s(w₂( · , · ))). For 0 < r < s < 1, the condition w₁, w₂ ∈ A_∞ suffices (A_∞ is the Muckenhoupt condition over rectangles). For 0 < r < 1 < s < ∞, it suffices that w₁ ∈ A_∞ and w₂ ∈ As. For 1 < r < s = ∞, we assume that the weights are of the form w_i(z₁, z₂) = a_i(z₁)ui(z₁, z₂)b_i(z₂), and then the following conditions suffice: u₁ ∈ A_p, u₂ ∈ A₁, \( {u}_2^p{u}_1\in {\mathrm{A}}_{\infty } \) , and log a_i, log b_i ∈ BMO. The last statement generalizes the previously known result for the case of u_i ≡ 1, i = 1, 2. Finally, for r = 1, s = ∞, the conditions w₁, w₂ ∈ A₁ and w₁w₂ ∈ A_∞ suffice.

Assume that σ > 0, r, μ ???? ℕ, μ ≥ r + 1, r is odd, p ???? [1,+∞], and \( f\kern0.5em \in \kern0.5em {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). We construct linear operators X_σ,r,μ whose values are splines of degree μ and of minimal defect with knots \( \frac{k\pi}{\sigma },k\in \mathrm{\mathbb{Z}} \), such that

\( {\displaystyle \begin{array}{l}{\left\Vert f-{X}_{\sigma, r,u}(f)\right\Vert}_p\le {\left(\frac{\pi }{\sigma}\right)}^r\left\{\frac{A_r,0}{2}\left.{\upomega}_1\right|{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p+\sum \limits_{v=1}^{u-r-1}{A}_{r,v}{\omega}_v{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p\right\}\\ {}\kern9em +{\left(\frac{\pi }{\sigma}\right)}^r\left(\frac{{\mathcal{K}}_r}{\pi^r}-\sum \limits_{v=0}^{u-r-1}{2}^v{A}_{r,v}\right){2}^{r-\mu }{\omega}_{\mu -r}{\left({f}^{(r)},\frac{\pi }{\sigma}\right)}_p,\end{array}} \) where for p = 1, . . . ,+∞, the constants cannot be reduced on the class \( {W}_p^{(r)}\left(\mathrm{\mathbb{R}}\right) \). Here \( {\mathcal{K}}_r=\frac{4}{\pi}\sum \limits_{l=0}^{\infty}\frac{{\left(-1\right)}^{l\left(r+1\right)}}{{\left(2l+1\right)}^{r+1}} \) are the Favard constants, the constants A_r,ν are constructed explicitly, and ω_v is a modulus of continuity of order ν. As a corollary, we get the sharp Jackson type inequality

\( {\left\Vert f-{X}_{\sigma, r,\mu }(f)\right\Vert}_p\le \frac{{\mathcal{K}}_r}{2{\sigma}^r}{\omega}_1{\left({f}^{(r)},\frac{\uppi}{\sigma}\right)}_p. \)

The paper is devoted to the problem of finding the exact constant \( {W}_2^{\ast } \) in the inequality ‖f‖ ≤ K ⋅ ω₂(f, 1) for bounded functions f with the property

\( \underset{k}{\overset{k+1}{\int }}f(x) dx=0,\kern0.5em k\in \mathrm{\mathbb{Z}}. \)

Our approach allows us to reduce the known range for the desired constant as well as the set of functions involved in the extremal problem for finding the constant in question. It is shown that \( {W}_2^{\ast } \) also turns out to be the exact constant in a related Jackson–Stechkin type inequality.

A generalization of some recent results of D. M. Stolyarov and the authors is proved.

Let \( {\left\{{\psi}_{j,k}\right\}}_{\left(j,k\right)\in {\mathrm{\mathbb{Z}}}^2} \) and \( {\left\{{\tilde{\psi}}_{j,k}\right\}}_{\left(j,k\right)\in {\mathrm{\mathbb{Z}}}^2} \) be dual wavelet frames in L₂(ℝ), let η be an even, bounded, decreasing on [0, ∞) function such that

\( \underset{0}{\overset{\infty }{\int }}\eta (x)\log \left(1+x\right) dx<\infty, \)

and let |ψ(x)|, \( \left|\tilde{\psi}(x)\right|\le \eta (x) \). Then the series \( \sum \limits_{j,k\in \mathrm{\mathbb{Z}}}\left(f,{\tilde{\psi}}_{j,k}\right){\psi}_{j,k} \) converges unconditionally in L_p(ℝ), 1 < p < ∞.

The results of a recent paper by A. V. Vasin, S. V. Kislyakov, and the author on the relationship between the local boundary smoothness of an analytic function and local boundary smoothness of its modulus are extended to the case of generalized pointwise Hölder type conditions of order between one and two.

We prove that if a function f is holomorphic in the polydisk ????ⁿ, n ≥ 2, f is continuous in \( \overline{{\mathbb{D}}^n} \), f(z) ≠ 0, z ∈ ????ⁿ, and |f| belongs to the α-Hölder class, 0 < α < 1, on the boundary of ????ⁿ, then f belongs to the \( \left(\frac{\alpha }{2}-\varepsilon \right) \)-Hölder class on \( \overline{{\mathbb{D}}^n} \) for any ε > 0.