Geodesic
The Gelfand and Bernstein widths of some classes of analytic functions. II
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 134-140 
O. G. Parfenov. The Gelfand and Bernstein widths of some classes of analytic functions. II. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 24, Tome 232 (1996), pp. 134-140. http://geodesic.mathdoc.fr/item/ZNSL_1996_232_a10/
@article{ZNSL_1996_232_a10,
     author = {O. G. Parfenov},
     title = {The {Gelfand} and {Bernstein} widths of some classes of analytic {functions.~II}},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {134--140},
     year = {1996},
     volume = {232},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1996_232_a10/}
}
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The Gelfand widths of the unit ball of $H^2(\nu)$ (the weighted Hardy space) with respect to the metric of the space $L_\infty(T_r)$ are considered ($T_r$ being the circle of radius $r$ centered at the origin), as well as the Bernstein widths of the unit ball of $H^\infty$ with respect to the metric of the space $L_2(T_r,\mu)$. The asymptotic formulas for the widths in the question are established for arbitrary measures $\nu,\mu$. Bibl. 5 titles.