Asymptotically optimal cubature formulas on manifolds for prefixed weights

We propose a new extension of the proof of the Korevaar-Meyers conjecture by Bondarenko, Radchenko and Viazovska for cubature formulas with prefixed weights (we fix different weights for different points) on algebraic and Riemannian manifolds.

[1]  L. Brandolini,et al.  On a sharp lemma of Cassels and Montgomery on manifolds , 2020, 2003.09339.

[2]  S. L. Sobolev,et al.  Theory of Cubature Formulas , 1997 .

[3]  M. Spivak.  A comprehensive introduction to differential geometry , 1979 .

[4]  J. Ortega-Cerdà,et al.  Sampling of real multivariate polynomials and pluripotential theory , 2015, 1509.00956.

[5]  Wenbo Dai,et al.  A new constructing approach for a weighted topology of wireless sensor networks based on local-world theory for the Internet of Things (IOT) , 2012, Comput. Math. Appl..

[6]  Bianca Gariboldi,et al.  Optimal asymptotic bounds for designs on manifolds , 2018, Analysis & PDE.

[7]  Y. Chikuse.  Statistics on special manifolds , 2003 .

[8]  J. Seidel,et al.  Spherical codes and designs , 1977 .

[9]  J. Korevaar,et al.  Chebyshev-type quadrature on multidimensional domains , 1994 .

[10]  S. Lang.  Introduction to Differentiable Manifolds , 1964 .

[11]  Hrushikesh Narhar Mhaskar,et al.  A Quadrature Formula for Diffusion Polynomials Corresponding to a Generalized Heat Kernel , 2010 .

[12]  Characterization of algebraic curves by Chebyshev quadrature , 1998 .

[13]  Giacomo Gigante,et al.  Diameter Bounded Equal Measure Partitions of Ahlfors Regular Metric Measure Spaces , 2015, Discrete & Computational Geometry.

[14]  S. Zelditch,et al.  Eigenfunctions of the Laplacian on a Riemannian manifold , 2017 .

[15]  Ronald Cools,et al.  An encyclopaedia of cubature formulas , 2003, J. Complex..

[16]  Hrushikesh Narhar Mhaskar,et al.  Diffusion polynomial frames on metric measure spaces , 2008 .

[17]  Andriy Bondarenko,et al.  Optimal asymptotic bounds for spherical designs , 2010, 1009.4407.

[18]  Joaquim Ortega-Cerdà,et al.  Asymptotically optimal designs on compact algebraic manifolds , 2016, 1612.06729.

[19]  Anna Breger,et al.  Quasi Monte Carlo Integration and Kernel-Based Function Approximation on Grassmannians , 2016, 1605.09165.

[20]  Vladimir N. Temlyakov,et al.  Cubature formulas, discrepancy, and nonlinear approximation , 2003, J. Complex..

[21]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[22]  Hrushikesh Narhar Mhaskar,et al.  Marcinkiewicz-Zygmund measures on manifolds , 2010, J. Complex..

[23]  John M. Lee.  Introduction to Smooth Manifolds , 2002 .