Quasiwidths and optimization of methods of mixed approximation of multidimensional singular integrals with kernels of hilbert type

We consider the problem of application of mixed methods to the construction of algorithms, optimal in accuracy, for the calculation of multidimensional singular integrals with Hilbert-type kernels. We propose a method for the optimization of cubature formulas for singular integrals with Hilbert-type kernels based on the theory of quasiwidths.     

Sharp estimates of the error of interpolation by bilinear splines for some classes of functions

On a two-point homogeneous space X, we consider the problem of describing the set of continuous functions having zero integrals over all spheres enclosing the given ball. We obtain the solution of this problem and its generalizations for an annular domain in X. By way of applications, we prove new uniqueness theorems for functions with zero spherical means.     

Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>

We find the exact values of the n-widths for the classes of periodic differentiable functions in L ₂[0, 2π] satisfying the constraint

Best polynomial approximations in L 2 of classes of 2π-periodic functions and exact values of their widths

We find sharp inequalities between the best approximations of periodic differentiable functions by trigonometric polynomials and moduli of continuity ofmth order in the space L ₂ as well as present their applications. For some classes of functions defined by these moduli of continuity, we calculate the exact values of n-widths in L ₂.     

Best polynomial approximations and the widths of function classes in L 2

Sharp Jackson-Stechkin type inequalities in which the modulus of continuity of mth order of functions is defined via the Steklov function are obtained. For the classes of functions defined by these moduli of continuity, exact values of various n-widths are derived.     

On the renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation

We study the problem of renewal of a solution of the Dirichlet boundary-value problem for a biharmonic equation on the basis of the known information about the boundary function. The obtained estimates of renewal error are unimprovable in certain cases. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 8, pp. 1147–1151, August, 1998.