Adaptive sampling recovery of functions with higher mixed regularity

We tensorize the Faber spline system from [14] to prove sequence space isomorphisms for multivariate function spaces with higher mixed regularity. The respective basis coefficients are local linear combinations of discrete function values similar as for the classical Faber Schauder system. This allows for a sparse representation of the function using a truncated series expansion by only storing discrete (finite) set of function values. The set of nodes where the function values are taken depends on the respective function in a non-linear way. Indeed, if we choose the basis functions adaptively it requires significantly less function values to represent the initial function up to accuracy $\varepsilon>0$ (say in $L_\infty$) compared to hyperbolic cross projections. In addition, due to the higher regularity of the Faber splines we overcome the (mixed) smoothness restriction $r<2$ and benefit from higher mixed regularity of the function. As a byproduct we present the solution of Problem 3.13 in the Triebel monograph [46] for the multivariate setting.

[1]  V. N. Temlyakov,et al.  Constructive Sparse Trigonometric Approximation for Functions with Small Mixed Smoothness , 2015, 1503.00282.

[2]  D. B. Bazarkhanov,et al.  Nonlinear approximations of classes of periodic functions of many variables , 2014 .

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

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

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

[6]  Dinh Dung.  Asymptotic orders of optimal non-linear approximations , 2001 .

[7]  Winfried Sickel,et al.  Tensor products of Sobolev-Besov spaces and applications to approximation from the hyperbolic cross , 2009, J. Approx. Theory.

[8]  A S Romanyuk.  Best $ M$-term trigonometric approximations of Besov classes of periodic functions of several variables , 2003 .

[9]  The best trigonometric approximations and the Kolmogorov diameters of the Besov classes of functions of many variables , 1993 .

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

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

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

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

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

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

[16]  Van Kien Nguyen,et al.  Change of Variable in Spaces of Mixed Smoothness and Numerical Integration of Multivariate Functions on the Unit Cube , 2015 .

[17]  Serhii A. Stasyuk.  Best m-term trigonometric approximation of periodic functions of several variables from Nikol'skii-Besov classes for small smoothness , 2014, J. Approx. Theory.

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

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

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

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

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

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

[24]  Winfried Sickel,et al.  Best m-term aproximation and tensor product of Sobolev and Besov spaces-the case of non-compact embeddings , 2010 .

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

[26]  Rajula Srivastava,et al.  Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces , 2020, ArXiv.

[27]  George G. Lorentz,et al.  Constructive Approximation , 1993, Grundlehren der mathematischen Wissenschaften.

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

[29]  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.

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

[31]  V. Temlyakov,et al.  Greedy Algorithms with Regard to Multivariate Systems with Special Structure , 1997 .

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

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

[34]  Ahmed E. Radwan,et al.  δβ-I APPROXIMATION SPACES , 2017 .

[35]  Dinh Dung,et al.  Continuous Algorithms in n-Term Approximation and Non-Linear Widths , 2000 .

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

[37]  Tino Ullrich,et al.  A higher order Faber spline basis for sampling discretization of functions , 2019, J. Approx. Theory.

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

[39]  R. DeVore,et al.  Nonlinear approximation , 1998, Acta Numerica.

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

[41]  Winfried Sickel,et al.  Best m-Term Approximation and Sobolev–Besov Spaces of Dominating Mixed Smoothness—the Case of Compact Embeddings , 2012 .

[42]  Winfried Sickel,et al.  Best m-term approximation and Lizorkin-Triebel spaces , 2011, J. Approx. Theory.

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

[44]  Vladimir Temlyakov,et al.  Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness , 2014, 1412.8647.

[45]  Best trigonometric and bilinear approximations for the Besov classes of functions of many variables , 1995 .

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

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

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

[49]  A. S. Romanyuk,et al.  Asymptotic estimates for the best trigonometric and bilinear approximations of classes of functions of several variables , 2010 .