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I. A. Sheipak

Publications and source records attributed to I. A. Sheipak.

11 recordsLinked to original sources
arXiv2026

On Sharp Estimates of Derivatives of Even Order

The norms of embedding operators of Sobolev spaces $\Wo^n_2[0;1]\hookrightarrow\Wo^k_\infty[0;1]$ ($0\leqslant k\leqslant n-1$) are considered. The least possible quantities $A^2_{n,k}(x)$ in the inequalities $|f^{(k)}(x)|^2\leqslant A^2_{n,k}(x)\|f^{(n)}\|^2_{L_2[0;1]}$ are studied. On the basis of the relations between the $A^2_{n,k}(x)$ and the antiderivatives of the Legendre polynomials, the properties of the maxima of the functions $A^2_{n,k}(x)$ are established. It is shown that, for all~$k$, the global maximum of the function $A^2_{n,k}$ on the closed interval $[0;1]$ is the maximum point nearest to the midpoint of the interval; in particular, for even~$k$, $x=1/2$ is such a point. For the parameter~$k$ of even order, an explicit formula for the norms of the embedding operators is obtained.

math.FA↗
arXiv2022

Exact estimates of high-order derivatives in Sobolev spaces

The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of the $L_{p'}[0;1]$ norm minimization for $Q^{(n)}_{n,k}$ ($1/ p+1/p'=1$) with the problem of the best estimates of derivatives of $y^{(k)}(a)\leqslant A_{n,k,p}(a)\|y^{(n)}\|_{L_p[0;1]}$, and also with the problem of finding the exact embedding constants of the Sobolev space $\mathring{W}^n_p[0;1]$ into the space $\mathring{W}^k_\infty[0;1]$, $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$. Exact embedding constants are found for $k=n-1$ and $p=\infty$, as well as for all $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$ and $p=1$.

math.FA↗
arXiv2011

On the Neumann problem for Sturm-Liouville equation with self-similar Cantor type weight

Sturm-Liouville problem with generalized derivative of self-similar Cantor type function as a weight is considered. Under Neumann and mixed boundary conditions the oscillating properties of the eigenfunctions are studied. The spectral asymptotics are made more precise then in previous papers. Namely, it is shown that for known asymptotics $N(λ)=λ^D\cdot [s(\lnλ)+o(1)]$ the function $s$ is a product of decreasing exponent and nondecreasing purely singular function (and hence it is not constant).

math.SP↗
arXiv2009

On spectrum of Jacobi operator with exponentially increasing matrix elements

The class of three-diagonal Jacobi matrix with exponentially increasing elements is considered. Under some assumptions the matrix corresponds to unbounded self-adjoint operator in the weighted space. The weight depends on elements of the matrix and in some cases can arise indefinite metric. We proved that eigenvalue problem for this operator is equivalent to the eigenvalue problem of Sturm--Liouville operator with discrete self-similar weight. The asymptotic formulas for eigenvalues are obtained. These formulas differ for cases of definite and indefinite metrics.

math.FA↗