Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

By K. Atkinson, Weimin Han. 2012

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AN EXTENSION OF A THEOREM OF SCHOENBERG TO PRODUCTS OF SPHERES (2015, background): “We suggest the classical reference [20] for the first topic and [2, 7, 11, 16] for the other.”
Depth Separation for Neural Networks (2017, unknown): “[1]) dμd(x) = Γ( d2 ) √ πΓ( d−1 2 ) (1 − x2) d−3 2 dx.”
Deep Neural Tangent Kernel and Laplace Kernel Have the Same RKHS (2020, background): “Interested readers are referred to [5] for spherical harmonics on the unit sphere.”
Towards Understanding the Spectral Bias of Deep Learning (2019, background): “For more detials, see Bach (2017); Bietti and Mairal (2019); Frye and Efthimiou (2012); Atkinson and Han (2012) for references.”
Breaking the Curse of Dimensionality with Convex Neural Networks (2014, background): “…all derivatives of the same order of g are bounded by a constant that only depends on d and α. Precise notions of differentiability may be defined on the sphere, using the manifold structure (see, e.g., Absil et al., 2009) or through polar coordinates (see, e.g., Atkinson and Han, 2012, Chapter 3).”