Approximation of Convolution Classes by Fourier Sums. New Results

We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.     

Best n-Term Approximations by Λ-Methods in the Spaces S ϕ p

We determine the exact values of the upper bounds of n-term approximations of q-ellipsoids by -methods in the spaces S ^q in the metrics of the spaces S ^p .     

Approximation of Cauchy-Type Integrals

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of -integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded rth derivatives (here, r is a natural number).On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of -integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.