We construct deep ReLU neural networks to approximate functions in dilated shift-invariant spaces generated by a continuous function with compact support and study the approximation rates with respect to the number of neurons . The network construction is based on the bit extraction and data fitting capacity of deep neural networks. Combining with existing results of approximation from shift-invariant spaces, we are able to estimate the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces  . We also give lower bounds of the L^p ( 0, 1 ^d \le \le  ) approximation error for Sobolev spaces, which show that our construction  is asymptotically optimal up to a logarithm factor.
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces , which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated with respect to the width and depth of neural networks . The network construction is based on the bit extraction and data-fitting capacity of deep neural networks. As applications of our main results, the approximation rates of classical function spaces such as Sobolev spaces and Besov spaces are obtained . We also give lower bounds of the L^p ( 1 \le p\le \infty ) approximation error for Sobolev spaces, which show that our construction of neural network is asymptotically optimal up to a logarithmic factor.
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construct
study the expressive power of
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to approximate
for approximating
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generated by a continuous function with compact support and study the approximation rates
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number of neurons
width and depth of neural networks
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data fitting
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Combining with existing results of approximation from shift-invariant spaces, we are able to estimate
As applications of our main results,
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are obtained
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0,
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^d
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p
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\infty
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of neural network
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logarithm
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