Universal Sampling Discretization

Let XN be an N -dimensional subspace of L2 functions on a probability space (Ω, μ) spanned by a uniformly bounded Riesz basis ΦN . Given an integer 1 ≤ v ≤ N and an exponent 1 ≤ q ≤ 2, we obtain universal discretization for integral norms Lq(Ω, μ) of functions from the collection of all subspaces ofXN spanned by v elements of ΦN with the number m of required points satisfying m ≪ v(logN)2(log v)2. This last bound on m is much better than previously known bounds which are quadratic in v. Our proof uses a conditional theorem on universal sampling discretization, and an inequality of entropy numbers in terms of greedy approximation with respect to dictionaries.

[1]  G. Schechtman.  More on embedding subspaces of $L_p$ in $l^n_r$ , 1987 .

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

[3]  R. DeVore,et al.  Nonlinear Approximation by Trigonometric Sums , 1995 .

[4]  Vladimir N. Temlyakov,et al.  Hyperbolic Cross Approximation , 2016, 1601.03978.

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

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

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

[8]  A. Pajor,et al.  Subspaces of small codimension of finite-dimensional Banach spaces , 1986 .

[9]  Feng Dai,et al.  Entropy numbers and Marcinkiewicz-type discretization , 2021 .

[10]  Vladimir N. Temlyakov,et al.  On the entropy numbers of the mixed smoothness function classes , 2016, J. Approx. Theory.

[11]  B. Carl.  Entropy numbers, s-numbers, and eigenvalue problems , 1981 .

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

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

[14]  C. Schütt.  Entropy numbers of diagonal operators between symmetric Banach spaces , 1984 .

[15]  G. Schechtman.  Tight embedding of subspaces of "L"p in "l n,p" for even "p" , 2011 .

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

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

[18]  E. S. Belinskii.  Decomposition theorems and approximation by a “floating" system of exponentials , 1998 .

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

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

[21]  M. Rudelson.  Almost orthogonal submatrices of an orthogonal matrix , 1996, math/9606213.

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

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

[24]  Aicke Hinrichs,et al.  Entropy numbers of embeddings of Schatten classes , 2016, 1612.08105.

[25]  Vladimir N. Temlyakov,et al.  An inequality for the entropy numbers and its application , 2013, J. Approx. Theory.