Approximation Theory and Harmonic Analysis on Spheres and Balls

By Feng Dai, Yuan Xu. 2013

579 citations recorded by the source corpus.

Subfields: Mechanics of Materials, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e500c7becbd2cd6511ebebaa853fd460febf2371

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In Statistical and Nonlinear Physics

Citations in the corpus

Adaptive Bayesian Procedures Using Random Series Priors (2014, unknown): “t of basis functions. The following examples illustrate the use of the theorem. Example 1 (Fourier trigonometric series). For a function w0 ∈ Cα(0,1), the best approximation has the error e(J) ≍ J−α [Dai and Xu, 2013]. Then the rate calculation proceeds in the following way: (2.16) implies J¯−α n . ”
Spectral bias and task-model alignment explain generalization in kernel regression and infinitely wide neural networks (2020, background): “The spherical parts depend on the radial distances of the data points ∣∣ x ∣∣ , whereas the angular components admit a decomposition in terms of spherical harmonics of the unit vectors Y km ð ^ x Þ , where k is the degree and m is the order of the harmonic 50 .”
Spectrum Dependent Learning Curves in Kernel Regression and Wide Neural Networks (2020, background): “Dot product kernels admit an expansion in terms of Gegenbauer polynomials { Q k } , which form a complete and orthonormal basis for the uniform measure on the sphere (Dai & Xu, 2013).”
On Learning Over-parameterized Neural Networks: A Functional Approximation Prospective (2019, background): “It holds that For each (cid:96) ∈ N , there exists a set of orthonormal basis { Y (cid:96),i : i = 1 , . . . , N (cid:96) } for H (cid:96) w. r. t. the uniform distribution ρ that can be written in terms of C ( λ ) in [DX13, Theorem 1.5.1] as This is known as the addition theorem .”
Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere (2016, unknown): “where Υk,d is an index set specified through the Appendix A2 (OS), Yk,ν,d : S → C are the normalized hyperspherical harmonics (Dai and Xu, 2013), and where the set of all ξk,ν(t) forms a countable infinite sequence of Gaussian processes, with zero mean and Eξk,ν(t)ξk′,ν′(t) = δk,k′δν,ν′gk(t − t′), t”
Irregular sampling of band-limited functions on the sphere (2014, unknown): “For example, see [1] and the references therein.”
Minimax Estimation of Functionals of Discrete Distributions (2014, unknown): “on simple polytopes4, balls, and spheres, and it remained open until Totik [160] generalized the results to general polytopes. For results in balls and spheres, the readers are referred to Dai and Xu [161]. We still know little about regimes other than polytopes, balls, and spheres. We would like to”
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.”
On Sobolev orthogonal polynomials (2014, unknown): “For properties of harmonic polynomials and their relations to orthogonal polynomials, we refer to [28, 34] and references therein.”
Positive $$L^p$$Lp-Bounded Dunkl-Type Generalized Translation Operator and Its Applications (2017, background): “l weight on Rd and Sd−1 were proved; see, e.g., the papers by C.F. Dunkl [14, 15, 16], M. Ro¨sler [41, 42, 43, 44], M.F.E. de Jeu [24, 25], K. Trim`eche [53, 54], Y. Xu [55, 56], and the recent works [1, 11, 12, 19, 20]. Date: March 21, 2017. 1991 Mathematics Subject Classiﬁcation. 42B10, 33C45, 33C”