This paper studies how efficiently deep ReLU neural networks can approximate and learn smooth functions. When the error is measured in $L^p([0,1]^d)$ norm and the approximator is a network with width $W$ and depth $L$, recent works have proven the supper approximation rate $\mathcal{O}((WL)^{-2s/d})$ for Besov space $\mathcal{B}^s_{q,r}([0,1]^d)$ under the Sobolev embedding condition $s/d>1/q-1/p$. In order to overcome the curse of dimensionality in this rate, we extent this result to anisotropic and mixed smooth function classes. We establish the approximation rate $\mathcal{O}((WL)^{-2\tilde{s}})$ for anisotropic Besov space $\mathcal{B}^{\boldsymbol{s}}_{q,r}([0,1]^d)$ with anisotropic smoothness $\boldsymbol{s}=(s_1,\dots,s_d)$ under the embedding condition $\tilde{s} > 1/q-1/p$, where the mean smoothness $\tilde{s} = (\sum_{i=1}^d s_i^{-1})^{-1}$. For mixed smooth Besov space $\mathcal{MB}^s_{q,r}([0,1]^d)$ with mixed smoothness $s>1/q-1/p$, we show that the approximation rate $\mathcal{O}((WL)^{-2s})$ holds up to logarithmic factors. Using these results, we also derive approximation bounds for the composition of anisotropic Besov functions. As an application, it is shown that deep ReLU neural networks can achieve minimax optimal rates up to logarithmic factors for a wide range of smooth function classes.

Deep learning has shown remarkable effectiveness in high-dimensional approximation problems, particularly in scientific computing, inverse problems, and operator learning (Han et al., 2018; Adcock et al., 2022; Beck et al., 2023). In many such settings, the ReLUs activation σs(t) = max{0,t}s (s N0) is especially relevant because it yields piecewisepolynomial representations that are well suited to smooth targets and derivative-sensitive tasks (Yang and Zhou, 2025; He et al., 2024).

We present a systematic recipe for translating ReLU approximation results to softmax Transformers1. Given a constructive ReLU approximator for a target, we construct an explicit softmax transformer with the same accuracy. The recipe applies to many common approximation targets and yields quantitative resource bounds beyond universal approximation statements. This matters because broad Universal Approximation Properties (UAP) still dominate Transformer approximation theory. For softmax Transformer, many universality results provide explicit constructions and quantitative resource bounds (e.g., parameters, depth, width...etc) [Yun et al., 2020, Kajitsuka and Sato, 2023, Takakura and Suzuki, 2023, Jiang and Li, 2024, Hu et al., 2025,

\citet{farrell2021deep} establish non-asymptotic high-probability bounds for general deep feedforward neural network (with rectified linear unit activation function) estimators, with \citet[Theorem 1]{farrell2021deep} achieving a suboptimal convergence rate for fully connected feedforward networks. The authors suggest that improved approximation of fully connected networks could yield sharper versions of \citet[Theorem 1]{farrell2021deep} without altering the theoretical framework. By deriving approximation bounds specifically for a narrower fully connected deep neural network, this note demonstrates that \citet[Theorem 1]{farrell2021deep} can be improved to achieve an optimal rate (up to a logarithmic factor). Furthermore, this note briefly shows that deep neural network estimators can mitigate the curse of dimensionality for functions with compositional structure and functions defined on manifolds.

A classical result of Carleman, based on the theory of quasianalytic functions, shows that polynomials are dense in $L^2(μ)$ for any $μ$ such that the moments $\int x^k dμ$ do not grow too rapidly as $k \to \infty$. In this work, we develop a fairly tight quantitative analogue of the underlying Denjoy-Carleman theorem via complex analysis, and show that this allows for nonasymptotic control of the rate of approximation by polynomials for any smooth function with polynomial growth at infinity. In many cases, this allows us to establish $L^2$ approximation-theoretic results for functions over general classes of distributions (e.g., multivariate sub-Gaussian or sub-exponential distributions) which were previously known only in special cases. As one application, we show that the Paley--Wiener class of functions bandlimited to $[-Ω,Ω]$ admits superexponential rates of approximation over all strictly sub-exponential distributions, which leads to a new characterization of the class. As another application, we solve an open problem recently posed by Chandrasekaran, Klivans, Kontonis, Meka and Stavropoulos on the smoothed analysis of learning, and also obtain quantitative improvements to their main results and applications.

We present simple, user-friendly bounds for the expected operator norm of a random kernel matrix under general conditions on the kernel function $k(\cdot,\cdot)$. Our approach uses decoupling results for U-statistics and the non-commutative Khintchine inequality to obtain upper and lower bounds depending only on scalar statistics of the kernel function and a ``correlation kernel'' matrix corresponding to $k(\cdot,\cdot)$. We then apply our method to provide new, tighter approximations for inner-product kernel matrices on general high-dimensional data, where the sample size and data dimension are polynomially related. Our method obtains simplified proofs of existing results that rely on the moment method and combinatorial arguments while also providing novel approximation results for the case of anisotropic Gaussian data. Finally, using similar techniques to our approximation result, we show a tighter lower bound on the bias of kernel regression with anisotropic Gaussian data.