Sampling, Marcinkiewicz-Zygmund Inequalities, Approximation, and Quadrature Rules

Given a sequence of Marcinkiewicz-Zygmund inequalities in $L^2$, we derive approximation theorems and quadrature rules. The derivation is completely elementary and requires only the definition of Marcinkiewicz-Zygmund inequality, Sobolev spaces, and the solution of least square problems.

[1]  Johann S. Brauchart,et al.  Numerical Integration over Spheres of Arbitrary Dimension , 2007 .

[2]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[3]  C. Choirat,et al.  Quadrature rules and distribution of points on manifolds , 2010, 1012.5409.

[4]  Ben Adcock,et al.  A Generalized Sampling Theorem for Stable Reconstructions in Arbitrary Bases , 2010, 1007.1852.

[5]  L. Hörmander.  The Analysis of Linear Partial Differential Operators III , 2007 .

[6]  Paul Neval,et al.  Ge´za Freud, orthogonal polynomials and Christoffel functions. A case study , 1986 .

[7]  V. Temlyakov,et al.  Integral norm discretization and related problems , 2018, Russian Mathematical Surveys.

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

[9]  H. Mhaskar,et al.  Approximate Quadrature Measures on Data-Defined Spaces , 2016, 1612.02368.

[10]  Ian H. Sloan,et al.  Cubature over the sphere S2 in Sobolev spaces of arbitrary order , 2006, J. Approx. Theory.

[11]  Doron S. Lubinsky,et al.  Marcinkiewicz-Zygmund Inequalities: Methods and Results , 1998 .

[12]  B. Pridhnani,et al.  Sufficient Conditions for Sampling and Interpolation on the Sphere , 2013, 1302.6730.

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

[14]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[15]  K. Gröchenig.  A discrete theory of irregular sampling , 1993 .

[16]  Giuseppe Mastroianni,et al.  Weighted Polynomial Inequalities with Doubling and A∞ Weights , 2000 .

[17]  Hans G. Feichtinger,et al.  Theory and practice of irregular sampling , 2021, Wavelets.

[18]  Joaquim Ortega-Cerda,et al.  Beurling–Landauʼs density on compact manifolds , 2011, 1105.2501.

[19]  Daan Huybrechs,et al.  Stable high-order quadrature rules with equidistant points , 2009, J. Comput. Appl. Math..

[20]  Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics , 2006, math/0611503.

[21]  William W. L. Chen,et al.  Discrepancy and Numerical Integration on Metric Measure Spaces , 2013, 1308.6775.

[22]  Ben Adcock,et al.  Beyond Consistent Reconstructions: Optimality and Sharp Bounds for Generalized Sampling, and Application to the Uniform Resampling Problem , 2013, SIAM J. Math. Anal..

[23]  L. Hörmander,et al.  The spectral function of an elliptic operator , 1968 .

[25]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

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

[27]  Wenchang Sun,et al.  Reconstruction of Band-Limited Functions from Local Averages , 2002 .

[28]  G. Gigante,et al.  Trigonometric approximation and a general form of the Erd\H{o}s Tur\'{a}n inequality , 2010, 1001.0948.

[29]  Lloyd N. Trefethen,et al.  Trigonometric Interpolation and Quadrature in Perturbed Points , 2016, SIAM J. Numer. Anal..

[30]  Doron S. Lubinsky,et al.  On Converse Marcinkiewicz—Zygmund Inequalities in Lp,p>1 , 1999 .

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

[32]  Karlheinz Gröchenig.  Non-Uniform Sampling in Higher Dimensions: From Trigonometric Polynomials to Bandlimited Functions , 2001 .

[33]  Lloyd N. Trefethen,et al.  The Exponentially Convergent Trapezoidal Rule , 2014, SIAM Rev..

[34]  A. Cohen,et al.  Optimal weighted least-squares methods , 2016, 1608.00512.

[35]  Ian H. Sloan,et al.  QMC designs: Optimal order Quasi Monte Carlo integration schemes on the sphere , 2012, Math. Comput..

[36]  S. Smale,et al.  Shannon sampling and function reconstruction from point values , 2004 .

[37]  Gabriele Steidl,et al.  Fast Fourier Transforms for Nonequispaced Data: A Tutorial , 2001 .

[38]  Jordi Saludes,et al.  Marcinkiewicz-Zygmund inequalities , 2007, J. Approx. Theory.

[39]  Karlheinz Gröchenig,et al.  Random Sampling of Multivariate Trigonometric Polynomials , 2005, SIAM J. Math. Anal..

[40]  Ben Adcock,et al.  Optimal sampling strategies for multivariate function approximation on general domains , 2019, SIAM J. Math. Data Sci..

[41]  T. Strohmer,et al.  Efficient numerical methods in non-uniform sampling theory , 1995 .

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

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

[44]  W. Sickel,et al.  Marcinkiewicz-Zygmund-Type Inequalities Trigonometric Interpolation on Non-Uniform Grids and Unconditional Schauder Bases in Besov Spaces on the Torus , 1997 .

[45]  J. Romero,et al.  Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates , 2016, Applied and Computational Harmonic Analysis.

[46]  Karlheinz Gröchenig,et al.  Irregular sampling, Toeplitz matrices, and the approximation of entire functions of exponential type , 1999, Math. Comput..

[47]  Isaac Z. Pesenson,et al.  Shannon Sampling and Weak Weyl’s Law on Compact Riemannian Manifolds , 2017, Springer Proceedings in Mathematics & Statistics.

[48]  J. Benedetto,et al.  Irregular sampling and the theory of frames, I , 1990 .

[49]  Hrushikesh Narhar Mhaskar,et al.  Spherical Marcinkiewicz-Zygmund inequalities and positive quadrature , 2001, Math. Comput..

[50]  L. Gårding.  On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators , 1953 .

[51]  Tamás Erdélyi,et al.  Notes on inequalities with doubling weights , 1999 .

[52]  Hans G. Feichtinger,et al.  Geometric Space–Frequency Analysis on Manifolds , 2015, 1512.08668.

[53]  M. Shubin.  Pseudodifferential Operators and Spectral Theory , 1987 .

[54]  Martin Ehler,et al.  Optimal Monte Carlo integration on closed manifolds , 2017, Statistics and Computing.

[55]  Albert Cohen,et al.  Sequential Sampling for Optimal Weighted Least Squares Approximations in Hierarchical Spaces , 2018, SIAM J. Math. Data Sci..

[56]  Joaquim Ortega-Cerdà,et al.  Equidistribution of Fekete Points on the Sphere , 2008, 0808.1202.