An Image Processing Problem for Computer-Synthesized Full-Color Holograms

The article considers relief computation for computer-synthesized holograms creating full-color images. The propagation of radiation from the optical element to the focal plane is described by scalar wave theory and is considered both in the long-focus approximation and in the geometrical optics approximation. The phase function specifying the optical element is from the set of admissible phase functions defined by purely technological constraints. A composite phase element is constructed in the geometrical optics approximation. The algorithms rely on numerical solution of the inverse problem in the long-focus approximation and on the analytical form of the phase function in the geometrical optics approximation. The optical elements generated by the proposed procedure have been manufactured by electron-beam lithography. Their comparison has shown that the composite phase element is much more efficient. Translated from Prikladnaya Matematika i Informatika, No. 3, pp. 65–70, 1999.     

Hybrid polynomial approximation to higher derivatives of rational curves

In this paper, we extend the results published in JCAM volume 214 pp. 163-174 in 2008. Based on the bound estimates of higher derivatives of both Bernstein basis functions and rational Bézier curves, we prove that for any given rational Bézier curve, if the convergence condition of the corresponding hybrid polynomial approximation is satisfied, then not only the l-th (l=1,2,3) derivatives of its hybrid polynomial approximation curve uniformly converge to the corresponding derivatives of the rational Bézier curve, but also this conclusion is tenable in the case of any order derivative. This result can expand the area of applications of hybrid polynomial approximation to rational curves in geometric design and geometric computation.     

The singular values of the imbedding operators of some classes of analytic functions of several variables

We discuss three results. The first exhibits the order of decrease of the s-values as a function of the C R-dimension of a compact set on which we approximate the class of analytic functions being studied. The second is an asymptotic formula for the case when the domain of analyticity and the compact set are Reinhart domains. The third is the computation of the s-values of a special operator that is of interest for approximation theory on one-dimensional manifolds. Bibliography: 6 titles.Translated fromProblemy Matematicheskogo Analiza, No. 12, 1992, pp. 115–127.     

Methods for determining node location in asymptotically uniform spline approximation

Equations are derived for node location in spline approximation of analytical functions such that the weighted errors in the L _t8 norm are close on each piece of the spline. There are no restrictions on the spline pieces: they may be in the form of polynomials, rational polynomials, or nonlinear expressions. The equations are solved by numerical methods which differ by the volume of computation and accuracy of the results. Some numerical examples are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 3–13, 1988.     

Fast computation of the field of a vertical magnetic dipole above a layered medium

We consider the direct problem of aeroelectrosounding for a layered medium. The specific feature of this problem is that its solution is representable as a Bessel transform of a function ƒ of a certain type. We have developed and implemented a method for fast computation of the Bessel transform of such functions. The method relies on the approximation of ƒ by a linear combination of exponentials with prespecified exponents. The Bessel transform of the original function ƒ is computed analytically as the Bessel transform of the approximating linear combination. Numerical computations have been carried out. __________ Translated from Prikladnaya Matematika i Informatika, No. 26, pp. 39–48, 2007.     

Computing the Approximation Error for Neural Networks with Weights Varying on Fixed Directions

We obtain a sharp lower bound estimate for the approximation error of a continuous function by single hidden layer neural networks with a continuous activation function and weights varying on two fixed directions. We show that for a certain class of activation functions this lower bound estimate turns into equality. The obtained result provides us with a method for direct computation of the approximation error. As an application, we give a formula, which can be used to compute instantly the approximation error for a class of functions having second order partial derivatives.     

Asymptotics of approximation of Ψ-differentiable functions of many variables

We investigate the approximation characteristics of classes of Ψ-differentiable functions of many variables introduced by Stepanets. We present the asymptotics of approximation of functions of these classes. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 8, pp. 1051–1057, August, 2008.     

Approximation of classes of smooth functions of several variables

The paper deals with imbedding theorems and harmonic approximation for classes of periodic functions of several variables. Necessary and sufficient conditions for imbedding are found for various classes of smooth functions. The paper contains asymptotic estimates for fractional order derivatives of multidimensional Dirichlet and Favard kernels, for Fourier best approximation and for the Kolmogorov diameter of smooth function classes.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 10, pp. 207–226, 1984.     

A new rational approximation technique based on transformational high dimensional model representation

In this work, a new rational approximation scheme, based on the recently developed Transformational High Dimensional Model Representation (THDMR) approximation method is developed. As an initial step to the construction of a rational approximation for multivariate functions via THDMR, this paper focuses on the general theoretical background of the method and gives explicit formulae for the computation of such approximants. The performance of the technique is shown by several examples both in univariate and bivariate cases.     

Polynomial Approximations on Closed Subsets of Elliptic Curves

Approximation of Holder functions on subsets of elliptic curves using polynomials is considered. An equivalent problem of approximation on the complex plane with the use of doubly periodic functions is introduced. A direct approximation theorem is proved. Bibliography: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 178–187.     

Basic properties of the problem of the best simultaneous approximation of several elements

For the problem of the best approximation of several elements with respect to the maximum of convex-concave fractional functions with additional restrictions, we establish duality relations and criteria for the element of the best approximation. Kamenets-Podol'skii Pedagogical Institute, Kamenets-Podol'sk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1183–1193, September, 1996.     

An alternative computational method for the approximation of random functions

Summary This paper shows that a computational procedure for approximation of random functions can be accomplished using purely linear programming techniques. This contrasts with previous results which use a twostage approach for the computation, one of which requires linear programming techniques. Computational results are given.     

Successive approximation by spaces of local functions

One considers interpolation and approximation by local functions on a nonuniform net and stable algorithms for their successive construction. One introduces the concept of H-stability of a family of interpolations and one determines sufficient conditions for H-stability. One constructs nonhomogeneous spaces of local functions which realize an a priori approximation of a given order; the basis functions of these spaces have in a certain sense a minimal support and possess interpolation properties.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 1l1, pp. 31–51, 1981.     

Asymptotics of small spacings for subsonic nonstationary streamlined flow of a thin airfoil close to a solid boundary

In an asymptotic approximation of small spacings an analytic solution to the problem on harmonic oscillations of a thin airfoil which is moving with a subsonic velocity near a solid plane boundary is given. Results of a computation of the lifting force are given.Translated from Dinamicheskie Sistemy, No. 7, pp. 48–53, 1988.     

Lipschitz condition for the operator of best uniform approximation at points of uniqueness

The operator of best uniform approximation of real, continuous functions by elements of a finite space is considered. It is shown that the Lipschitz condition for the operator of best approximation by generalized polynomials is satisfied for each function having a characteristic set of maximal dimension.Translated from Matematicheskie Metody i Fiziko-Mekhanicheskie Polya, No. 25, pp. 17–19, 1987.     

Uniform estimates for monotone approximation

Uniform estimates are obtained for the monotone approximation of the functions from the generalized Babenko classes.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 6, pp. 785–790, June, 1993.     

Best linear methods of approximation and optimal orthonormal systems of the Hardy space

We construct the best linear methods of approximation for functions of the Hardy space H _p on compact subsets of the unit disk. It is shown that the Takenaka-Malmquist systems are optimal systems of functions orthonormal on the unit circle for the construction of the best linear methods of approximation. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 636–646, May, 2008.     

Approximation of Functions in the Spaces <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>(ℝ) by Means of Shifts and Contractions of One Function

Some approximation problems for functions of class L_p are considered. Bibliography: 4 titles.__________Translated from Problemy Matematicheskogo Analiza, No. 30, 2005, pp. 47–59.     

Approximation of fractional-order integrals by algebraic polynomials. II

We investigate the approximation of functions that are fractional-order integrals of bounded functions by algebraic polynomials. Dnepropetrovsk University, Dnepropetrovsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 940–951, July, 1999.