A higher order Faber spline basis for sampling discretization of functions

Abstract This paper is devoted to the question of constructing a higher order Faber spline basis for the sampling discretization of functions with higher regularity than Lipschitz. The basis constructed in this paper has similar properties as the piecewise linear classical Faber–Schauder basis (Faber, 1908) except for the compactness of the support. Although the new basis functions are supported on the real line they are very well localized (exponentially decaying) and the main parts are concentrated on a segment. This construction gives a complete answer to Problem 3.13 in Triebel’s monograph (Triebel, 2012) by extending the classical Faber basis to higher orders. Roughly, the crucial idea to obtain a higher order Faber spline basis is to apply Taylor’s remainder formula to the dual Chui–Wang wavelets. As a first step we explicitly determine these dual wavelets which may be of independent interest. Using this new basis we provide sampling characterizations for Besov and Triebel–Lizorkin spaces and overcome the smoothness restriction coming from the classical piecewise linear Faber–Schauder system. This basis is unconditional and coefficient functionals are computed from discrete function values similar as for the Faber–Schauder situation.

[1]  Dinh Dung,et al.  Continuous algorithms in adaptive sampling recovery , 2013, J. Approx. Theory.

[2]  Jan Vyb'iral,et al.  Non-smooth atomic decompositions, traces on Lipschitz domains, and pointwise multipliers in function spaces , 2012, 1201.2280.

[3]  Multiple Haar Basis and m-term Approximations for Functions from the Besov Classes. I , 2016 .

[4]  Über die Existenz von Schauderbasen in Sobolev-Besov-Räumen. Isomorphiebeziehungen , 1973 .

[5]  C. Chui,et al.  On compactly supported spline wavelets and a duality principle , 1992 .

[6]  Tino Ullrich,et al.  Lower Bounds for Haar Projections: Deterministic Examples , 2015, 1511.01470.

[7]  Jürgen Prestin,et al.  On an orthogonal bivariate trigonometric Schauder basis for the space of continuous functions , 2019, J. Approx. Theory.

[8]  Hans Triebel.  Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration , 2012 .

[9]  Dinh Dung,et al.  Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation , 2012, Found. Comput. Math..

[10]  D. Dung.  B-Spline Quasi-Interpolation Sampling Representation and Sampling Recovery in Sobolev Spaces of Mixed Smoothness , 2016, 1603.01937.

[11]  Jan Vybíral.  Function spaces with dominating mixed smoothness , 2006 .

[12]  Dinh Dung,et al.  B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness , 2010, J. Complex..

[13]  Franklin Richards,et al.  The Lebesgue constants for cardinal spline interpolation , 1975 .

[14]  Jianzhong Wang,et al.  CUBIC SPLINE WAVELET BASES OF SOBOLEV SPACES AND MULTILEVEL INTERPOLATION , 1996 .

[15]  A. Haar.  Zur Theorie der orthogonalen Funktionensysteme , 1910 .

[16]  Z. Ciesielski.  Spline Bases in Function Spaces , 1975 .

[17]  Jürgen Prestin,et al.  On a constructive representation of an orthogonal trigonometric Schauder basis for C 2π , 2001 .

[18]  H. Bungartz,et al.  Sparse grids , 2004, Acta Numerica.

[19]  Aicke Hinrichs,et al.  Optimal quasi-Monte Carlo rules on order 2 digital nets for the numerical integration of multivariate periodic functions , 2016, Numerische Mathematik.

[20]  P. Oswald,et al.  Multilevel Finite Element Riesz Bases in Sobolev Spaces , 1998 .

[21]  G. Bourdaud.  Ondelettes et espaces de Besov , 1995 .

[22]  Mario Ullrich,et al.  The Role of Frolov's Cubature Formula for Functions with Bounded Mixed Derivative , 2015, SIAM J. Numer. Anal..

[23]  Multiple Haar Basis and its Properties , 2016 .

[24]  T. Ullrich.  Local Mean Characterization of Besov-Triebel-Lizorkin Type Spaces with Dominating Mixed Smoothness on Rectangular Domains , 2008 .

[25]  Dinh Dũng,et al.  Dimension-dependent error estimates for sampling recovery on Smolyak grids based on B-spline quasi-interpolation , 2020, J. Approx. Theory.

[26]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[27]  H. Triebel,et al.  Topics in Fourier Analysis and Function Spaces , 1987 .

[28]  Tino Ullrich,et al.  Haar projection numbers and failure of unconditional convergence in Sobolev spaces , 2015, Mathematische Zeitschrift.

[29]  H. Triebel.  Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration , 2010 .

[30]  Z. Ciesielski.  Properties of Realizations of Random Fields , 1980 .

[31]  M. Schäfer,et al.  Hyperbolic Wavelet Analysis of Classical Isotropic and Anisotropic Besov–Sobolev Spaces , 2019, Journal of Fourier Analysis and Applications.

[32]  Dinh Dng.  Full length article: Continuous algorithms in adaptive sampling recovery , 2013 .

[33]  H. Triebel,et al.  Intrinsic atomic characterizations of function spaces on domains , 1996 .

[34]  G. Kyriazis,et al.  Decomposition systems for function spaces , 2003 .

[35]  Tino Ullrich,et al.  Basis Properties of the Haar System in Limiting Besov Spaces , 2021, Geometric Aspects of Harmonic Analysis.

[36]  Hans Triebel,et al.  Function Spaces with Dominating Mixed Smoothness , 2019 .

[37]  Discrete Characterization of Besov Spaces and Its Applications to Stochastics , 1997 .

[38]  Tino Ullrich,et al.  Function Spaces with Dominating Mixed Smoothness Characterization by Differences , 2006 .

[39]  B. Scharf.  Atomic representations in function spaces and applications to pointwise multipliers and diffeomorphisms, a new approach , 2011, 1111.6812.

[40]  Amara Lynn Graps,et al.  An introduction to wavelets , 1995 .

[41]  Winfried Sickel,et al.  SAMPLING THEORY AND FUNCTION SPACES , 2000 .

[42]  The Haar System as a Schauder Basis in Spaces of Hardy–Sobolev Type , 2016, 1609.08225.

[43]  Dinh Dng.  B-spline quasi-interpolant representations and sampling recovery of functions with mixed smoothness , 2011 .

[44]  P. Oswald.  Semiorthogonal linear prewavelets on irregular meshes , 2006 .

[45]  Dinh DźNg.  Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation , 2016 .

[46]  G. Faber.  Über stetige Funktionen , 1908 .

[47]  Janina Decker,et al.  A Mathematical Introduction To Wavelets , 2016 .

[48]  Jürgen Prestin,et al.  Characterization of Local Besov Spaces via Wavelet Basis Expansions , 2017, Front. Appl. Math. Stat..

[49]  Dinh Dung,et al.  Non-linear sampling recovery based on quasi-interpolant wavelet representations , 2009, Adv. Comput. Math..