Geodesic
Optimal rate of integration and $\varepsilon$-entropy of a~class of analytic functions
Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 3-10

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The author considers a class $F$ of analytic functions real in the interval $[-1,1]$ and bounded in the unit circle. As an estimate of the optimal quadrature error $R(n)$ over the class $F$ it is shown that $$ e^{\left(-2\sqrt2+\frac1{\sqrt2}\right)\pi\sqrt n}\le R(n)\le e{-\frac\pi{\sqrt2}n}. $$ With the additional condition that $\max\limits_{x\in[-1,1]}|f(x)|\le B$, an estimate is obtained for the $\varepsilon$-entropy $H_\varepsilon(F)$: $$ \frac8{27}\frac{(\ln2)^2}{\pi^2}\le\lim\frac{H_\varepsilon(F)}{(\log\frac1\varepsilon)^3}\le\frac2{\pi^2}(\ln2)^2. $$ 
@article{MZM_1973_14_1_a0,
     author = {B. D. Boyanov},
     title = {Optimal rate of integration and $\varepsilon$-entropy of a~class of analytic functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {3--10},
     publisher = {mathdoc},
     volume = {14},
     number = {1},
     year = {1973},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a0/}
}
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B. D. Boyanov. Optimal rate of integration and $\varepsilon$-entropy of a~class of analytic functions. Matematičeskie zametki, Tome 14 (1973) no. 1, pp. 3-10. http://geodesic.mathdoc.fr/item/MZM_1973_14_1_a0/