Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $\phi, \eta$ such that every formal conjugation $\hat{\sigma}$ (i.e. $\eta \circ \hat{\sigma} = \hat{\sigma} \circ \phi$) does not extend to the fixed points set $Fix (\phi)$ of $\phi$, meaning that it is not transversally formal (or semi-convergent) along $Fix (\phi)$. We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.Comment: 34 page

Divergence conditions for Riesz means of Rademacher functions

Some of the divergence conditions for Riesz means of Rademacher functions have been determined. A condition was provided as a criterion for at least one diverging sequence to be summable by the Riesz method. The Rademacher functions can be regarded as independent random variables on the specific probability space with the usual Lebesgue measure. To prove the divergence of a specific number sequence on a set of positive measures, an assertion on the means of Rademacher functions obtained by an arbitrary summation method was considered

Sharp estimates for functions of class W̊2 r(- 1,1)

A study was conducted to investigate sharp estimates for functions of class Sobolev spaces W̊r 2(-1, 1). The class Sobolev spaces consisted of all functions f : [-1, 1] ? R having absolutely continuous derivatives up to order r - 1. Calculation of exact constants in Kolmogorov-type inequalities for intermediate derivatives in various cases was considered in monographs. The zero boundary conditions on f are automatically satisfied at the left end, while at the right end (x = +1), they were equivalent to the family of orthogonality conditions

Erratum: Sharp estimates for functions of class W̊ 2 r (-1, 1) (Doklady Mathematics (2009) 80:2 (713-715))

[No abstract available

Sharp constants in one-dimensional Poincaré-type inequalities

[No abstract available

Sharp estimates for derivatives of functions in the Sobolev classes W̊2 r(-1, 1)

Explicit formulas are obtained for the maximum possible values of the derivatives f(k)(x), x ∈ (-1, 1), k ∈ {0,1,...,r - 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r - 1 at the points ±1 and are such that {double pipe}f(r){double pipe}L2(-1,1) ≤ 1. As a corollary, it is shown that the first eigenvalue λ1,r of the operator (-D2)r with these boundary conditions is √2 (2r)! (1 + O(1/r)), r → ∞. © 2010 Pleiades Publishing, Ltd

On existence conditions for sequences uniformly distributed with respect to Voronoi's methods

Necessary and sufficient conditions for the uniformly distributed sequences of complex numbers with respect to Voronoi and Riesz methods have been analyzed. These conditions are necessary and sufficient for the Voronoi and Riesz methods to be regular, for the convergence of the given sequence to a finite limit to imply the convergence of Voronoi and Riesz means, respectively, to the same limit. For a sequence of ones, the methods of Voronoi and Riesz coincide with the classical methods of Cesaro's arithmetic means. A weight sequence was constructed such that both necessary and sufficient conditions hold

Sharp estimates for derivatives of functions in the Sobolev classes W̊2 r(-1, 1)

Explicit formulas are obtained for the maximum possible values of the derivatives f(k)(x), x ∈ (-1, 1), k ∈ {0,1,...,r - 1}, for functions f that vanish together with their (absolutely continuous) derivatives of order up to ≤ r - 1 at the points ±1 and are such that {double pipe}f(r){double pipe}L2(-1,1) ≤ 1. As a corollary, it is shown that the first eigenvalue λ1,r of the operator (-D2)r with these boundary conditions is √2 (2r)! (1 + O(1/r)), r → ∞. © 2010 Pleiades Publishing, Ltd

On two-sided and asymptotic estimates for the norms of embedding operators of W̊2 n(-1, 1) into Lq(dμ)

Explicit upper and lower estimates are given for the norms of the operators of embedding of W̊2 n(-1, 1), n ∈ ℕ, in Lq(dμ), 0 < q < ∞. Conditions on the measure μ are obtained under which the ratio of the above estimates tends to 1 as n → ∞, and asymptotic formulas are presented for these norms in regular cases. As a corollary, an asymptotic formula (as n → ∞) is established for the minimum eigenvalues λ1, n, β, β > 0, of the boundary value problems (-d2/dx2)nu(x) = λ{pipe}x{pipe}β-1u(x), x ∈ (-1, 1), u(k)(±1) = 0, k ∈ {0, 1,..., n - 1}. © 2014 Pleiades Publishing, Ltd

Sharp estimates for functions of class W̊2 r(- 1,1)

A study was conducted to investigate sharp estimates for functions of class Sobolev spaces W̊r 2(-1, 1). The class Sobolev spaces consisted of all functions f : [-1, 1] ? R having absolutely continuous derivatives up to order r - 1. Calculation of exact constants in Kolmogorov-type inequalities for intermediate derivatives in various cases was considered in monographs. The zero boundary conditions on f are automatically satisfied at the left end, while at the right end (x = +1), they were equivalent to the family of orthogonality conditions