Best Approximations for the Cauchy Kernel on the Real Axis

Abstract

We compute the values of the best approximations for the Cauchy kernel on the real axis $ℝ$ by some subspaces from $L_q (ℝ)$. 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 $H_p,\; 2 ≤ p < ∞$.