We consider the optimal recovery problem of isotropic classes of r-th differentiable multivariate functions defined on (d), and obtain some asymptotically optimal results. It turns out that this optimal recovery probl...
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We consider the optimal recovery problem of isotropic classes of r-th differentiable multivariate functions defined on (d), and obtain some asymptotically optimal results. It turns out that this optimal recovery problem is intimately related to the optimal covering problem of (d) by equal balls in discrete geometry.
Questions dealing with the approximation of functions from the classes C (beta) (psi) H (alpha) by Poisson integrals are studied. The Kolmogorov-Nikol'skii problem for Poisson integrals for the classes C (beta) (p...
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Questions dealing with the approximation of functions from the classes C (beta) (psi) H (alpha) by Poisson integrals are studied. The Kolmogorov-Nikol'skii problem for Poisson integrals for the classes C (beta) (psi) H (alpha) is solved in the uniform metric.
Problems in the theory of approximation of functions on an arbitrary compact rank-one symmetric space M in the metric of L-p, 1 <= p <= infinity, are investigated. The approximating functions are generalized sph...
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Problems in the theory of approximation of functions on an arbitrary compact rank-one symmetric space M in the metric of L-p, 1 <= p <= infinity, are investigated. The approximating functions are generalized spherical polynomials, that is, linear combinations of eigenfunctions of the Beltrami-Laplace operator on M. Analogues of the direct, Jackson theorems are proved for the modulus of smoothness (of arbitrary order) constructed by using the operator of spherical averaging. It is established that the modulus, of smoothness and the K-functional constructed from the Sobolev-type space corresponding to the Beltrami-Laplace differential operator are equivalent.
approximation problems for functions on the half-line [0,+infinity) in a weighted L(p)-metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of s...
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approximation problems for functions on the half-line [0,+infinity) in a weighted L(p)-metric are studied with the use of Bessel generalized translation. A direct theorem of Jackson type is proven for the modulus of smoothness of arbitrary order which is constructed on the basis of Bessel generalized translation. Equivalence is stated between the modulus of smoothness and the K-functional constructed by the Sobolev space corresponding to the Bessel differential operator. A particular class of entire functions of exponential type is used for approximation. The problems under consideration are studied mostly by means of Fourier-Bessel harmonic analysis.
A brief overview of applications of Schoenberg's polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the fou...
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A brief overview of applications of Schoenberg's polynomial B-splines of odd degrees in mathematical statistics, computational mathematics, and statistical radio engineering is provided. Exact formulas for the found Schoenberg B-spline of 15th degree are presented. High-quality approximations of smooth functions with an infinite Fourier transform by functions with a finite Fourier transform are found.
In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the s...
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In this paper we shall introduce new constructions of approximate solutions of general linear partial differential equations with constant coefficients on the whole spaces, and establish fundamental estimates of the solutions depending on the inhomogeneous terms. This will be done by combining general ideas of the Tikhonov regularization and discretization of bounded linear operator equations on reproducing kernel Hilbert spaces. Furthermore, we will provide approximate solutions for the related inverse source problems.
The double-period method [1] uses special trigonometric series for approximation and extrapolation of nonperiodical functions. It has a number of advantages in comparison with other methods. In the previous work [2], ...
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On the basis of the resolvent of a simple differential operator, a method for finding approximations to continuous functions is constructed. In this method, both the approximated function and its approximations satisf...
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On the basis of the resolvent of a simple differential operator, a method for finding approximations to continuous functions is constructed. In this method, both the approximated function and its approximations satisfy the given integral boundary condition.
We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only o...
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We are concerned with the problem of uniform approximation of a continuous function of two variables by a product of continuous functions of one variable on some domain D. This problem have been examined so far only on a rectangular domain D = U x V, where U and V are compact sets. An algorithm to give a solution of this problem in the discrete case is available. We put forward an algorithm which in certain cases allows one to construct an approximate solution of the problem on a given domain (not necessarily rectangular). This approximate solution is built in the form of interpolating natural splines, which in turn are constructed by means of discrete approximation. Depending on the degree of the splines, the problem can be solved in classes of functions with appropriate degree of smoothness. Also, a double end estimate of the best approximation is obtained, and a method for improving both the solution and the double end estimate is proposed. approximation of functions by a product of continuous functions on the annulus and by a product of twice continuously differentiable functions on the domain bounded by the ellipse is considered by means of examples.
This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation ...
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This paper suggests a method of approximating the solution of minimization problems for convex functions of several variables under convex constraints is suggested. The main idea of this approach is the approximation of a convex function by a piecewise linear function, which results in replacing the problem of convex programming by a linear programming problem. To carry out such an approximation, the epigraph of a convex function is approximated by the projection of a polytope of greater dimension. In the first part of the paper, the problem is considered for functions of one variable. In this case, an algorithm for approximating the epigraph of a convex function by a polygon is presented, it is shown that this algorithm is optimal with respect to the number of vertices of the polygon, and exact bounds for this number are obtained. After this, using an induction procedure, the algorithm is generalized to certain classes of functions of several variables. Applying the suggested method, polynomial algorithms for an approximate calculation of the L(p)-norm of a matrix and of the minimum of the entropy function on a polytope are obtained.
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