On best uniform approximations by splines in the presence of restrictions on their derivatives

Translated from Matematicheskie Zametki, Vol. 50, No. 6, pp. 24–30, December, 1991.     

On additive inequalities for intermediate derivatives of functions given on a finite interval

We present a general scheme for deducing additive inequalities of Landau-Hadamard type. As a consequence, we prove several new inequalities for the norms of intermediate derivatives of functions given on a finite interval with an exact constant with the norm of a function.     

On Kolmogorov-type inequalities for fractional derivatives of functions of two variables

We prove a new exact Kolmogorov-type inequality estimating the norm of a mixed fractional-order derivative (in Marchaud's sense) of a function of two variables via the norm of the function and the norms of its partial derivatives of the first order. Translated from Ukrains'kyi Matematchnyi Zhurnal, Vol. 60, No. 6, pp. 837–842, June, 2008.     

On the best interval quadrature formulae for classes of differentiable periodic functions

In this paper we solve the problem about optimal interval quadrature formula for the class W^rF of differentiable periodic functions with rearrangement invariant set F of their derivatives of order r. We prove that the formula with equal coefficients and n node intervals having equidistant midpoints is optimal for considering classes. To this end a sharp inequality for antiderivatives of rearrangements of averaged monosplines is proved.     

BestL 1-approximations of classesW 1 r by Splines fromW 1 r

We obtain the exact values of the bestL ₁-approximations of the classesW ₁ ^r of periodic functions by periodic polynomial splines of degreer and defect 1 with equidistant knots that belong to the classW ₁ ^r .Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 10, pp. 1410–1413, October, 1994.     

Best approximation estimates resulting from the Chebyshev criterion

Translated from Matematicheskie Zametki, Vol. 49, No. 4, pp. 148–150, 1991.     

The effect of radiation on dye-doped polystyrene microspheres coated by a thin silver layer

The heat conduction equation is analyzed theoretically and numerically for the case of the effect of laser radiation on two-layer spherical particles. The internal heat sources and the temperature distribution are calculated for particles whose core of dye-doped polystyrene is coated by a deposited silver layer. The distribution of heat sources inside such a two-component microsphere is shown to depend significantly on the core radius and the shell thickness. Therefore, these parameters strongly affect the heating of particles. In particular, an increase in the thickness of the deposited silver layer may either reduce or strongly increase the heating time of two-layer microspheres.     

Sharp asymptotics of the <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> 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 \mathbbR^d{\mathbb{R}^d} . 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.     

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [?π,π) of the function sin(n + 1)t ? 2q sin nt, q ∈ ?, by the subspace of trigonometric polynomials of degree at most n ? 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.     

On Jackson-Type Inequalities for Functions Defined on a Sphere

We obtain exact estimates for the approximation of functions defined on a sphere in the metrics of C and L ₂ by linear methods of summation of Fourier series in spherical harmonics in the case where differential and difference properties of these functions are defined in the space L ₂. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 291–304, March, 2005.