In what follows, $\sigma>0$, $m,r\in\mathbb N$, $m\geqslant r$, ${\mathbf S}_{\sigma, m}$ is the space of splines of order~$m$ and minimal defect with nodes $\frac{j\pi}{\sigma}$ ($j\in\mathbb Z$), $A_{\sigma,m}(f)_{p}$ is the best approximation of a function~$f$ by the set ${\mathbf S}_{\sigma,m}$ in the space $L_p(\mathbb R)$. It is known that for $p=1,+\infty$

$$

\supl_{f\in W^{(r)}_{p}(\mathbb R)}

\frac{A_{\sigma,m}(f)_{p}}{\|f^{(r)}\|_{p}}=

\frac{{\mathcal K}_r}{\sigma^r}.\eqno(1)

$$

In this paper we construct linear operators ${\mathcal X}_{\sigma,r,m}$ with their values in ${\mathbf S}_{\sigma,m}$, such that for all $p\in[1,+\infty]$ and $f\in W_p^{(r)}(\mathbb R)$

$$

\|f-{\mathcal X}_{\sigma,r,m}(f)\|_{p}\leqslant

\frac{{\mathcal K}_r}{\sigma^r}\|f^{(r)}\|_p.

$$

So we establish the possibility to achieve the upper bounds in~(1) by linear methods of approximation, which was unknown before.

Some results of S. N. Bernstein about polynomials of least deviation from zero in some weighted spaces L p are generalized to entire functions of exponential type. Suppose that a function ρm belongs to the Cartwright class, is of type m, is positive on the real axis, and let σ ≥ m. Earlier, the authors constructed functions least deviating from zero among entire functions of type σ in the uniform and integral metrics on R with the weights ω = 1/ρm and ω = | · |/ρm. In this paper it is shown that these functions deviate least from zero in some other classes related to the function ρm and generalizing the Orlicz classes. In particular, the results are obtained for the spaces L p (ℝ), p < ∞, with the weight ω p for the same ω.

Results of Chebyshev and Bernstein about polynomials with the smallest deviation from zero in a weighted norm are extended to entire functions of exponential type. Suppose that a function \rho_m belongs to the Cartwright class, is of type m, and is positive on the real axis. Let \sigma\geqslant m. Functions that have the smallest deviation from zero among the entire functions of type \sigma are constructed in the uniform and integral metrics.

A definition of the category of Witt-correspondences over a field of characteristic different from 2 is given, the presheafs with Witt-transfers are introduced, and a series of general properties of these objects are established. In Theorem 1, the injectivity theorem is shown to be true for a homotopy invariant presheaf with Witt-transfers and for the local ring of a smooth variety. As a consequence, the injectivity theorem is proved for the Witt functor.

Rubio de Francia proved the one-sided Littlewood–Paley inequality for arbitrary intervals in *L**^p*, 2≤*p*<∞. In this article, such an inequality is proved for the Walsh system.

Let F ∈ S, S be the standard class of conformal mapping of the unit disk \mathbb{D}, and let . Suppose that there exist Jordan domains G_1 and G, G_1 ⊃ G and G_1 ⊃ G such that G ⊂ \mathbb{C}\f (\mathbb{D}), \partial{f}(\mathbb{D}) ∩ \partial{G}, contains a Dini-smoth arc \gamma, and G_1 ∩ \partial{f}(\mathbb{D}) (D) ∩ \partial{G}=\gamma. It is established that, in this case, for any r, 0 < r < 1 with F, does not maximize the expression Z |z|=r 1 |F ′(z)| 2 |dz| in the class S.

Metric aspects of the problem of ideals are studied. Let h be a function in the class H^{\infty}(\mathbb{D}) and f a vector-valued function in the class H^{\infty}(\mathbb{D}; E) , i.e., takes values in some lattice of sequences E. Suppose that |h(z)| \le \|f(z)\|_{E}^{\alpha} \le 1 for some parameter \alpha. The task is to find g a function in H^{\infty}(\mathbb{D}; E'), where E' is the order dual of E, such that \sum f_j g_j = h. Also it is necessary to control the value of \|g\|_{H^{\infty}(E')}. The classical case with E = l^2 was investigated by V. A. Tolokonnikov in 1981. Recently, the author managed to obtain a similar result for the space E=l^1. In this paper it is shown that the problem of ideals can be solved for any q-concave Banach lattice E with finite q; in particular E=l^p, with p \in [1,\infty] fits.

Generating functions for a fixed genus map and hypermap enumeration become rational after a simple explicit change of variables. Their numerators are polynomials with integral coefficients that obey a differential recursion, and the denominators are products of powers of explicit linear functions.

Normalized Laplacians and their perturbations by periodic potentials (Schrödinger operators) on periodic discrete graphs are treated. The spectrum of such an operator consists of an absolutely continuous part, which is the union of a finite number of nondegenerate bands, and a finite number of flat bands, i.e., eigenvalues of infinite multiplicity. Estimates for the Lebesgue measure of the spectrum in terms of geometric parameters of the graphs are obtained and it is shown that these estimates become identities for some graphs. Two-sided estimates are given for the lengths of the first spectral bands and for the effective masses at the bottom of the spectrum for the Laplace and Schrödinger operators. In particular, these estimates show that the first spectral band of the Schrödinger operators is nondegenerate.

The discrete spectrum of the Dirichlet problem for the Laplace operator on the union of two circular unit cylinders whose axes intersect at the right angle consists of a single eigenvalue. For the threshold value of the spectral parameter, this problem has no bounded solutions. When the angle between the axes reduces, the multiplicity of the discrete spectrum grows unboundedly.

The tropical arithmetic operations on R are defined as (a,b) -> min{a,b} and (a,b) -> a+b. We are interested in the concept of a semimodule, which is rather ill-behaved in tropical mathematics. In our paper we study the semimodules S in R^n having topological dimension two, and we show that any such S has always a finite weak dimension not exceeding n. For a fixed k, we construct a polynomial time algorithm deciding whether S is contained in some tropical semimodule of weak dimension k or not. The latter result provides a solution of a problem that has been open for eight years.

In the paper, the concept of a semimodule is discussed, which is rather ill-behaved in tropical mathematics. The semimodules S in R^n having topological dimension two are studied, and it is shown that any such S has a finite weak dimension not exceeding n. For a fixed k, a polynomial time algorithm is constructed that decides whether S is contained in some tropical semimodule of weak dimension k or not. This result provides a solution of a problem that has been open for eight years.

We consider a periodic Schrödinger operator H on a discrete periodic graph. Estimates of the discrete spectrum perturbed operator decreasing potential. In the case of a potential with a power-law the asymptotics at infinity is found asymptotics of the discrete spectrum of the operator for large coupling constant.

We prove a number of important results on the inner factors of analytic functions of variable smoothness in the closed disc