  We show expression rates and stability in Sobolev norms of deep feedforward
ReLU neural networks (NNs) in terms of the number of parameters defining the NN
for continuous, piecewise polynomial functions, on arbitrary, finite partitions
$\mathcal{T}$ of a bounded interval $(a,b)$. Novel constructions of ReLU NN
surrogates encoding function approximations in terms of Chebyshev polynomial
expansion coefficients are developed which require fewer neurons than previous
constructions. Chebyshev coefficients can be computed easily from the values of
the function in the Clenshaw--Curtis points using the inverse fast Fourier
transform. Bounds on expression rates and stability are obtained that are
superior to those of constructions based on ReLU NN emulations of monomials as
considered in [Opschoor, Petersen and Schwab, 2020] and [Montanelli, Yang and
Du, 2021]. All emulation bounds are explicit in terms of the (arbitrary)
partition of the interval, the target emulation accuracy and the polynomial
degree in each element of the partition. ReLU NN emulation error estimates are
provided for various classes of functions and norms, commonly encountered in
numerical analysis. In particular, we show exponential ReLU emulation rate
bounds for analytic functions with point singularities and develop an interface
between Chebfun approximations and constructive ReLU NN emulations.
