Department of Theory of functions

We construct a best linear method for the approximation of bounded harmonic functions on compact subsets of the unit disk. We show that a system of functions orthonormal on the unit circle and optimal for the construction of the best linear approximation method is a Takenaka-Malmquist system.

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

We show that the Lebesgue-Landau constants of linear methods for summation of Taylor series of functions holomorphic in a polydisk and in the unit ball from C m over triangular domains do not depend on the number m. On the basis of this fact, we find a relation between the complete and partial best approximations of holomorphic functions in a polydisk and in the unit ball from C

We construct a linear method {ie910-01} for the approximation (in the unit disk) of classes of holomorphic functions {ie910-02} that are the Hadamard convolutions of the unit balls of the Bergman space A p with reproducing kernels {ie910-03}. We give conditions for ψ under which the method {ie910-04} approximates the class {ie910-05} in the metrics of the Hardy space H s and the Bergman space A s , 1 ≤ s ≤ p, with an error that coincides in order with the value of the best approximation by algebraic polynomials.

A version of the mean-value theorem (formulas of finite increments) for analytic functions is proved. Take any function f(z) analytic in the disk I z I < r, r > 0, with f" (0) r 0. exist r o, 0 < r 0 < r, and a point ~ in the disk I zl < I z01 = I zl I for which for any points Zo and Zl UDC 517.5 Cinquini [1] proved that there f(z!)-f(Zo) = f'(~)(Zl-z0) (1)

We establish necessary and sufficient conditions under which a real-valued function from Lp(T), 1 ≤ p < ∞, is badly approximable by the Hardy subspace H 0 p := {f ∈ Hp : f (0) = 0}. In a number of cases, we obtain the exact values of the best approximations in the mean of functions holomorphic in the unit disk by functions holomorphic outside this disk. We use the obtained results for finding the exact values of the best polynomial approximations and n-widths of some classes of holomorphic functions. We establish necessary and sufficient conditions under which the generalized Bernstein inequality for algebraic polynomials on the unit circle is true.

We investigate the problem of approximation of functions f holomorphic in the unit disk by means A f r ρ, () as ρ → 1-. In terms of the error of approximation by these means, a constructive characteristic of classes of holomorphic functions H p r Lip α is given. The problem of the saturation of A f r ρ, () in the Hardy space H p is solved.

We consider functions of many complex variables that are holomorphic in a polydisk or in the upper half-plane. We give necessary and sufficient conditions under which a holomorphic function is a Cauchy-Stieltjes-type integral of a complex charge. We present several applications of this criterion to integral representations of certain classes of holomorphic functions.

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.

UDC 517.5 We establish conditions on the boundary of a bounded simply connected domain C under which the p-Faber series of an arbitrary function from the Smirnov space E p. /; 1 Ä p < 1; can be summed by the Abel-Poisson method on the boundary of the domain up to the limit values of the function itself in the metric of the space L p ./:

We determine the asymptotic equality for the upper bounds of deviations of generalized Zygmund sums Z n,ψ (f)(z) = f 0 + n−1 k=1 (1 − ψ n /ψ k) f k z k on the functional classes H ψφ p that are convolution of unit ball of the Hardy space H p with kernels ∞ k=0 ψ k+1 φ k+1 z k in case when ψ = {ψ k } ∞ k=1 are the moment sequence. We give necessary and sufficient conditions on the sequence φ = {φ k } ∞ k=1 under which the sums Z n,ψ (f) approximate the class H ψφ p with minimal possible error |ψ n |.

On concentric circles T = {z ∈ C : |z| = }, 0 ≤ < 1, we determine the exact values of the quantities of the best approximation of holomorphic functions of the Bergman class Ap, 2 ≤ p ≤ ∞, in the uniform metric by algebraic polynomials generated by linear methods of summation of Taylor series. For 1 ≤ p < 2, we establish exact order estimates for these quantities.

UDC 517.5 We compute the values of the best approximations for the Cauchy kernel on the real axis R by some subspaces from Lq(R). This result is applied to the evaluation of the sharp upper bounds for pointwise deviations of certain interpolation operators with interpolation nodes in the upper half plane and certain linear means of the Fourier series in the Takenaka-Malmquist system from the functions lying in the unit ball of the Hardy space Hp, 2  p < 1.