We prove a theorem concerning the approximation of multivariate functions by deep ReLU networks. We present new error estimates for which the curse of the dimensionality is lessened by establishing a connection with sparse grids. These results automatically apply to deep networks since deep networks include networks with a singe layer.2 A function σ : R → [0, 1] is said to be a sigmoid function if it is non-decreasing, lim x→−∞ σ(x) = 0 and lim x→+∞ σ(x) = 1, e.g., σ(x) = 1/(1 + e −x ). This does not include the ReLU activation function. Some density results for ReLU functions can be found in the review of Pinkus (e.g., [13, Prop. 3.7]).

We prove a theorem concerning the approximation of generalized bandlimited multivariate functions by deep ReLU networks for which the curse of the dimensionality is overcome. Our theorem is based on a result by Maurey and on the ability of deep ReLU networks to approximate Chebyshev polynomials and analytic functions efficiently.

Abstract. Algorithms and underlying mathematics are presented for numerical computation with periodic functions via approximations to machine precision by trigonometric polynomials, including the solution of linear and nonlinear periodic ordinary differential equations. Differences from the nonperiodic Chebyshev case are highlighted.

Intracellular symmetry breaking plays a key role in wide range of biological processes, both in single cells and in multicellular organisms. An important class of symmetry-breaking mechanisms relies on the cytoplasm/membrane redistribution of proteins that can autocatalytically promote their own recruitment to the plasma membrane. We present an analytical construction and a comprehensive parametric analysis of stable localized patterns in a reaction-diffusion model of such a mechanism in a spherical cell. The constructed patterns take the form of high-concentration patches localized into spherical caps, similar to the patterns observed in the studies of symmetry breaking in single cells and early embryos.