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Yuliya Babenko

Publications and source records attributed to Yuliya Babenko.

14 recordsLinked to original sources
arXiv2021

On the Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Polyá and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemmanian manifolds and derive the well-known Taikov and Hardy-Littlewood-Polyá inequalities for functions defined on $d$-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of other class. In addition, we establish sharp Solyar-type inequalities for unbounded closed operators with closed range.

math.FA↗
arXiv2017

Stechkin's problem for functions of a self-adjoint operator in a Hilbert space, Taikov-type inequalities and their applications

In this paper we solve the problem of approximating functionals $(φ(A)x, f)$ (where $φ(A)$ is some function of self-adjoint operator $A$) on the class of elements of a Hilbert space that is defined with the help of another function $ψ(A)$ of the operator $A$. In addition, we obtain a series of sharp Taikov-type additive inequalities that estimate $|(φ(A)x, f)|$ with the help of $\| ψ(A)x\|$ and $\| x\|$. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H${\rm{\ddot{o}}}$rmander theorem on comparison of operators in the case when operators are acting in a Hilbert space and are functions of a self-adjoint operator. As another application we obtain Taikov-type inequalities for functions of the operator $\frac1i \frac {d}{dt}$ in the spaces $L_2(\RR)$ and $L_2(\TT)$, as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials.

math.FA↗
arXiv2015

Inequalities of Hardy-Littlewood-Polya type for functions of operators and their applications

In this paper, we derive a generalized multiplicative Hardy-Littlewood-Polya type inequality, as well as several related additive inequalities, for functions of operators in Hilbert spaces. In addition, we find the modulus of continuity of a function of an operator on a class of elements defined with the help of another function of the operator. We then apply the results to solve the following problems: (i) the problem of approximating a function of an unbounded self-adjoint operator by bounded operators, (ii) the problem of best approximation of a certain class of elements from a Hilbert space by another class, and (iii) the problem of optimal recovery of an operator on a class of elements given with an error.

math.FA↗
arXiv2015

Optimal recovery of integral operators and its applications

In this paper we present the solution to the problem of recovering rather arbitrary integral operator based on incomplete information with error. We apply the main result to obtain optimal methods of recovery and compute the optimal error for the solutions to certain integral equations as well as boundary and initial value problems for various PDE's.

math.AP↗
arXiv2013

Exact asymptotics of the optimal Lp-error of asymmetric linear spline approximation

In this paper we study the best asymmetric (sometimes also called penalized or sign-sensitive) approximation in the metrics of the space $L_p$, $1\leqslant p\leqslant\infty$, of functions $f\in C^2\left([0,1]^2\right)$ with nonnegative Hessian by piecewise linear splines $s\in S(\triangle_N)$, generated by given triangulations $\triangle_N$ with $N$ elements. We find the exact asymptotic behavior of optimal (over triangulations $\triangle_N$ and splines $s\in S(\triangle_N)$ error of such approximation as $N\to \infty$.

math.NA↗
arXiv2013

Kolmogorov's Problem on the Class of Multiply Monotone Functions

In this paper we give necessary and sufficient conditions for the system of positive numbers $ M_{k_1}, M_{k_2},..., M_{k_{d}},$ $0\leq k_1<...<k_{d} {\leq} r$, to guarantee the existence of an $r$-monotone function defined on the negative half-line $\RR_-$ and such that $\|x^{(k_i)}\|_{\infty}=M_{k_i}, i=1,2,...,d$.

math.FA↗
arXiv2011

On the $L_p$-error of approximation of bivariate functions by harmonic splines

Interpolation by various types of splines is the standard procedure in many applications. In this paper we shall discuss harmonic spline "interpolation" (on the lines of a grid) as an alternative to polynomial spline interpolation (at vertices of a grid). We will discuss some advantages and drawbacks of this approach and present the asymptotics of the $L_p$-error for adaptive approximation by harmonic splines.

math.NA↗
arXiv2011

Exact asymptotics of the optimal $L_{p,\Omega}$-error of linear spline interpolation

In this paper we provide the exact asymptotics of the optimal weighted $L_p$-error, $0<p< \infty$, of linear spline interpolation of $C^2$ functions with positive Hessian. The full description of the behavior of the optimal error leads to the algorithm for construction of an asymptotically optimal sequence of triangulations. In addition, we compute the minimum of the $L_p$-error of linear interpolation of the function $x^2+y^2$ over all triangles of unit area for all $0<p<\infty$. This provides the exact constant in the asymptotics of the optimal error.

math.NA↗
arXiv2011

On one extremal property of a regular simplex

In this paper, we show that the $L_p$-error of asymmetric linear approximation of the quadratic function $Q({\mathbf x})=\sum_{j=1}^{d}x_j^2$ on simplices in $\RR^d$ of fixed volume is minimized on regular simplices.

math.NA↗
arXiv2011

Exact asymptotics of the uniform error of interpolation by multilinear splines

The question of adaptive mesh generation for approximation by splines has been studied for a number of years by various authors. The results have numerous applications in computational and discrete geometry, computer aided geometric design, finite element methods for numerical solutions of partial differential equations, image processing, and mesh generation for computer graphics, among others. In this paper we will investigate the questions regarding adaptive approximation of C2 functions with arbitrary but fixed throughout the domain signature by multilinear splines. In particular, we will study the asymptotic behavior of the optimal error of the weighted uniform approximation by interpolating and quasi-interpolating multilinear splines.

math.NA↗
arXiv2011

Sharp asymptotics of the Lp approximation error for interpolation on block partitions

Adaptive approximation (or interpolation) takes into account local variations in the behavior of the given function, adjusts the approximant depending on it, and hence yields the smaller error of approximation. The question of constructing optimal approximating spline for each function proved to be very hard. In fact, no polynomial time algorithm of adaptive spline approximation can be designed and no exact formula for the optimal error of approximation can be given. Therefore, the next natural question would be to study the asymptotic behavior of the error and construct asymptotically optimal sequences of partitions. In this paper we provide sharp asymptotic estimates for the error of interpolation by splines on block partitions in IRd. We consider various projection operators to define the interpolant and provide the analysis of the exact constant in the asymptotics as well as its explicit form in certain cases.

math.NA↗