We show that the discrete Hilbert transform and the discrete Kak–Hilbert transform are infinitesimal generators of one-parameter groups of operators in ${{\ell }^{2}}$.

A relatively long-standing problem concerning the best constants of approximation by entire functions of exponential type on the class $W^rH_{\omega}$ is solved. The central tool is a limit relation between the best constants of approximation by trigonometric polynomials and entire functions of exponential type on $W^rH_{\omega}$ and its periodic analogue.

The convergence behavior of best uniform rational approximations with numerator degree m and denominator degree n to the function |x|α, α > 0, on [-1, 1] is investigated. It is assumed that the indices (m, n) progress along a ray sequence in the lower triangle of the Walsh table, i.e. the sequence of indices {(m, n)} satisfies

In addition to the convergence behavior, the asymptotic distribution of poles and zeros of the approximants and the distribution of the extreme points of the error function on [-1, 1] will be studied. The results will be compared with those for paradiagonal sequences (m = n + 2[α/2]) and for sequences of best polynomial approximants.