Gel'fand widths of the unit ball of the Hardy class \(H^ p\) in weight spaces (Q1075561)

Let \(H^ p\), \(1\leq p\leq \infty\), be the classical Hardy class of order p, \(B_ p\) the unit disk in \(H^ p\), \(T_ r=[| z| =r]\), \(\mu\) a finite Borel measure on \(T_ r\) and \(L_ q(T_ r,\mu)\), \(1\leq q\leq \infty\), the usual space of Lebesgue integrable functions of order q. Suppose that \(d^ n=d^ n(B_ p(T_ r,\mu))\) denotes the n-th diameter in the sense of Gel'fand between the two indicated sets. The author proved that \[ \lim_{n\to \infty}d^ nr^{1-n- (1/q)}=(2\pi)^{(1/q)-(1/p)}(g(\mu))^{1/q}, \] where \(g(\mu)=\exp [(2\pi r)^{-1}\int_{T_ r}\ln p(z)| dz|]\).