  A three-hidden-layer neural network with super approximation power is
introduced. This network is built with the floor function ($\lfloor x\rfloor$),
the exponential function ($2^x$), the step function ($1_{x\geq 0}$), or their
compositions as the activation function in each neuron and hence we call such
networks as Floor-Exponential-Step (FLES) networks. For any width
hyper-parameter $N\in\mathbb{N}^+$, it is shown that FLES networks with width
$\max\{d,N\}$ and three hidden layers can uniformly approximate a H\"older
continuous function $f$ on $[0,1]^d$ with an exponential approximation rate
$3\lambda (2\sqrt{d})^{\alpha} 2^{-\alpha N}$, where $\alpha \in(0,1]$ and
$\lambda>0$ are the H\"older order and constant, respectively. More generally
for an arbitrary continuous function $f$ on $[0,1]^d$ with a modulus of
continuity $\omega_f(\cdot)$, the constructive approximation rate is
$2\omega_f(2\sqrt{d}){2^{-N}}+\omega_f(2\sqrt{d}\,2^{-N})$. Moreover, we extend
such a result to general bounded continuous functions on a bounded set
$E\subseteq\mathbb{R}^d$. As a consequence, this new class of networks
overcomes the curse of dimensionality in approximation power when the variation
of $\omega_f(r)$ as $r\rightarrow 0$ is moderate (e.g., $\omega_f(r)\lesssim
r^\alpha$ for H\"older continuous functions), since the major term to be
concerned in our approximation rate is essentially $\sqrt{d}$ times a function
of $N$ independent of $d$ within the modulus of continuity. Finally, we extend
our analysis to derive similar approximation results in the $L^p$-norm for
$p\in[1,\infty)$ via replacing Floor-Exponential-Step activation functions by
continuous activation functions.
