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

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

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

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 $N\to \infty$

Simultaneous Approximation of a Multivariate Function and its Derivatives by Multilinear Splines

In this paper we consider the approximation of a function by its interpolating multilinear spline and the approximation of its derivatives by the derivatives of the corresponding spline. We derive formulas for the uniform approximation error on classes of functions with moduli of continuity bounded above by certain majorants.Comment: 21 page

Stechkin\u27s 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 kψ(A)xk and kxk. We also present several applications of the obtained results. First, we find sharp constants in inequalities of the type used in H¨ormander 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 1 i d dt in the spaces L2(R) and L2(T), as well as for integrals with respect to spectral measures, defined with the help of classical orthogonal polynomials