Sampling Discretization of the Uniform Norm

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. We pay special attention to the case of trigonometric polynomials with frequencies from an arbitrary finite set with fixed cardinality. We give two different proofs of the fact that for any N -dimensional subspace of the space of continuous functions it is sufficient to use eCN sample points for an accurate upper bound for the uniform norm. Previous known results show that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. Also, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best m-term bilinear approximation of the Dirichlet kernel associated with the given subspace. We illustrate the application of our technique on the example of trigonometric polynomials.

[1]  V. Temlyakov,et al.  Remez-Type and Nikol’skii-Type Inequalities: General Relations and the Hyperbolic Cross Polynomials , 2017 .

[2]  Toni Volkmer,et al.  Worst case recovery guarantees for least squares approximation using random samples , 2019, ArXiv.

[3]  V. N. Temlyakov,et al.  The Marcinkiewicz-Type Discretization Theorems , 2017, Constructive Approximation.

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

[5]  Vladimir N. Temlyakov,et al.  On optimal recovery in L2 , 2021, J. Complex..

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

[7]  Mario Ullrich,et al.  Function values are enough for L2-approximation: Part II , 2020, J. Complex..

[8]  Vladimir N. Temlyakov,et al.  Universal discretization , 2017, J. Complex..

[9]  C. D. Boor,et al.  MULTIVARIATE APPROXIMATION , 1986 .

[10]  V. Temlyakov,et al.  On optimal recovery in L 2 . , 2020, 2010.03103.

[11]  M. Ruiz Espejo.  Sampling , 2013, Encyclopedic Dictionary of Archaeology.

[12]  Nikhil Srivastava,et al.  Twice-ramanujan sparsifiers , 2008, STOC '09.

[13]  Tino Ullrich,et al.  A new upper bound for sampling numbers , 2020, ArXiv.

[14]  P. Heywood.  Trigonometric Series , 1968, Nature.

[15]  Vladimir Temlyakov,et al.  The Volume Estimates and Their Applications , 2003 .

[16]  E. Novak.  Deterministic and Stochastic Error Bounds in Numerical Analysis , 1988 .

[17]  Gideon Schechtman,et al.  Factorizations of natural embeddings of $l^{n}_{p}$ into $L_{r}$, I , 1988 .

[18]  Karlheinz Grochenig,et al.  Sampling, Marcinkiewicz–Zygmund inequalities, approximation, and quadrature rules , 2019, Journal of Approximation Theory.

[19]  V. Temlyakov,et al.  Observations on discretization of trigonometric polynomials with given spectrum , 2018, Russian Mathematical Surveys.

[20]  Stefan Kunis,et al.  Efficient Reconstruction of Functions on the Sphere from Scattered Data , 2007 .

[21]  Tino Ullrich,et al.  A note on sampling recovery of multivariate functions in the uniform norm , 2021, SIAM J. Numer. Anal..

[22]  Alexander Ulanovskii,et al.  Exponential frames on unbounded sets , 2014 .

[23]  V. Temlyakov,et al.  Entropy numbers and Marcinkiewicz-type discretization theorem , 2020, 2001.10636.

[24]  D. Spielman,et al.  Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem , 2013, 1306.3969.

[25]  Vladimir N. Temlyakov,et al.  On Approximate Recovery of Functions with Bounded Mixed Derivative , 1993, J. Complex..

[26]  J. Lindenstrauss,et al.  Approximation of zonoids by zonotopes , 1989 .

[27]  Vladimir Temlyakov,et al.  CAMBRIDGE MONOGRAPHS ON APPLIED AND COMPUTATIONAL MATHEMATICS , 2022 .

[28]  V. N. Temlyakov,et al.  The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials , 2017, 1702.01617.

[29]  F. Dai,et al.  Positive Cubature Formulas and Marcinkiewicz–Zygmund Inequalities on Spherical Caps , 2007, math/0703768.

[30]  Adam Krzyzak,et al.  A Distribution-Free Theory of Nonparametric Regression , 2002, Springer series in statistics.

[31]  D. R. Lewis.  Finite dimensional subspaces of $L_{p}$ , 1978 .