= Entropy of a smooth periodic function ball
{title2=$c\varepsilon^{-n/p}\le H\le C\varepsilon^{-n/p}\log(1/\varepsilon)$}
For the unit ball of $C^p(\mathbb T^n)$ measured in the uniform metric, $c\varepsilon^{-n/p}\le H(\varepsilon)\le C\varepsilon^{-n/p}\log(1/\varepsilon)$. The lower bound uses disjoint <bump functions> of spatial scale $h$, height proportional to $h^p$ and independently chosen zero/one coefficients: there are $\exp(c h^{-n})$ functions separated at scale $h^p$. For the upper bound, <Multivariable Jackson approximation> gives a <trigonometric polynomial> of degree $O(\varepsilon^{-1/p})$. Its $O(\varepsilon^{-n/p})$ coefficients are bounded and can be rounded with coefficient accuracy proportional to $\varepsilon$ divided by the coefficient count. Counting those rounded coefficient arrays supplies the logarithmic factor. The continuous-function unit ball without derivative bounds has infinite entropy for $\varepsilon<1/2$, by an infinite sequence of disjoint height-one bumps.
Back to article page
