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Haar Projection Numbers and Failure of Unconditional Convergence

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Haar Projection Numbers and Failure of Unconditional Convergence Haar projection numbers and failure of unconditional convergence Tino Ullrich Hausdorff-Zentrum f¨urMathematik / Institut f¨urNumerische Simulation Universit¨atBonn Strobl, June 2016 Institute for Numerical Simulation Rheinische Friedrich-Wilhelms-Universität Bonn Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 1 / 24 Based on three recent papers... A. Seeger, T. Ullrich, Haar projection numbers and failure of unconditional convergence in Sobolev spaces, Math. Zeitschrift, to appear. A. Seeger, T. Ullrich, Lower bounds for Haar projections, deterministic examples, Constr. Approx., to appear. G. Garrig´os,A. Seeger, T. Ullrich, The Haar system as a Schauder basis in Triebel-Lizorkin spaces, in preparation. Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 2 / 24 The Haar system h(x) 1 x 1 1 2 −1 The univariate Haar wavelet system j hj;k (x) := h(2 x − k) ; j 2 N0; k 2 Z : j=2 The system f2 hj;k gj;k is an orthonormal basis in L2(R) Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 3 / 24 Simple building blocks... with applications in Image and signal processing Probability (Brownian motion, small ball problems) (multivariate) Approximation theory Discrepancy theory ... Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 4 / 24 Faber-Schauder system j vj;k := v(2 · −k) ; j 2 N0; k 2 Z : Faber = R Haar h(t) v(x) 1 1 2 R x v(x)= 0 h(t) dt x 1 1 t −−−−−−−−−! 1 1 2 2 −1 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 5 / 24 Faber = R Haar h(t) v(x) 1 1 2 R x v(x)= 0 h(t) dt x 1 1 t −−−−−−−−−! 1 1 2 2 −1 Faber-Schauder system j vj;k := v(2 · −k) ; j 2 N0; k 2 Z : Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 5 / 24 The Faber-Schauder basis Faber 1908: Univariate hat functions: decomposition of f 2 C(T) 1 2j −1 1 X X f (x) = f (0) − d (f )v 2 j;k j;k j=0 k=0 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 5 / 24 The Faber-Schauder basis Faber 1908: Univariate hat functions: decomposition of f 2 C(T) 1 2j −1 1 X X f (x) = f (0) − d (f )v 2 j;k j;k j=0 k=0 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 6 / 24 Multivariate Faber-Schauder basis Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 7 / 24 Sparse grid: Optimal recovery of functions Sparse grid, d = 2, N = 256 1 3 4 1 2 1 4 0 0 1 1 3 1 4 2 4 Point clouds Digital net: Optimal cubature Digital net, d = 2, N = 256 1 3 4 1 2 1 4 0 0 1 1 3 1 4 2 4 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 8 / 24 Sparse grid, d = 2, N = 256 1 3 4 1 2 1 4 0 0 1 1 3 1 4 2 4 Point clouds Digital net: Sparse grid: Optimal cubature Optimal recovery of functions Digital net, d = 2, N = 256 1 3 4 1 2 1 4 0 0 1 1 3 1 4 2 4 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 8 / 24 s Haar basis in the Sobolev space Hp(R)? Sharp conditions? s What about Triebel-Lizorkin spaces Fp;q(R)? s s Note Fp;2(R) = Hp if 1 < p < 1 100 years Haar and Faber basis 1908 Faber, Uber¨ stetige Funktionen, Math. Ann. 66 1910 Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 1910 Faber, Uber¨ die Orthogonalfunktionen des Herrn Haar, DMV Jahresber. 1928 Schauder, Eine Eigenschaft der Haarschen Orthog.-systeme, Math. Z. 28 1932 Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 1937 Marcinkiewicz, Quelques th´eor. sur les s´er.orthogonales, Ann. Sci. Math. 1972 Ciesielski, Spline bases in function spaces, Proc. conf. Appr. Theory 1973 Triebel, Uber¨ die Existenz von Schauderbasen in Sobolev-R¨aumen 1978 Triebel, On Haar bases in Besov spaces, Serdica 4 2010 Triebel Discrepancy, numerical integration, and hyperbolic cross approximation, EMS Z¨urich Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 9 / 24 100 years Haar and Faber basis 1908 Faber, Uber¨ stetige Funktionen, Math. Ann. 66 1910 Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 1910 Faber, Uber¨ die Orthogonalfunktionen des Herrn Haar, DMV Jahresber. 1928 Schauder, Eine Eigenschaft der Haarschen Orthog.-systeme, Math. Z. 28 1932 Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 1937 Marcinkiewicz, Quelques th´eor. sur les s´er.orthogonales, Ann. Sci. Math. 1972 Ciesielski, Spline bases in function spaces, Proc. conf. Appr. Theory 1973 Triebel, Uber¨ die Existenz von Schauderbasen in Sobolev-R¨aumen 1978 Triebel, On Haar bases in Besov spaces, Serdica 4 2010 Triebel Discrepancy, numerical integration, and hyperbolic cross approximation, EMS Z¨urich s Haar basis in the Sobolev space Hp(R)? Sharp conditions? s What about Triebel-Lizorkin spaces Fp;q(R)? s s Note Fp;2(R) = Hp if 1 < p < 1 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 9 / 24 An unconditional basis in Lp ikx In contrast to the trigonometric system fe gk2Z the Haar wavelet system is an unconditional basis in Lp for p 6= 2 Theorem (Marcinkiewicz '37, Paley '32) Let 1 < p < 1. (i) The Haar system fhj;k gj;k2Z is an unconditional basis in Lp(R). (ii) There exist constants 0 < c < C such that 2 1=2 X X j ckf kp ≤ 2 hf ; hj;k iχj;k ≤ Ckf kp p j2Z k2Z The statement does neither hold in case p = 1 nor p = 1 ! Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 10 / 24 Littlewood-Paley, ( k )k \local mean kernels" 1 1=2 X ks2 2 kf kHs 2 j k ∗ f (·)j p p k=0 Triebel-Lizorkin spaces, 0 < p; q ≤ 1, s 2 R 1 1=q X ksq q kf kF s := 2 j k ∗ f (·)j p;q p k=0 s The Haar function belongs to Hp if s < 1=p s Fractional smoothness: the Sobolev space Hp Sobolev regularity, 1 < p < 1, s 2 R −1 2 s=2 kf k s := kF [(1 + jξj ) Ff ]k Hp(R) p Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 11 / 24 Triebel-Lizorkin spaces, 0 < p; q ≤ 1, s 2 R 1 1=q X ksq q kf kF s := 2 j k ∗ f (·)j p;q p k=0 s The Haar function belongs to Hp if s < 1=p s Fractional smoothness: the Sobolev space Hp Sobolev regularity, 1 < p < 1, s 2 R −1 2 s=2 kf k s := kF [(1 + jξj ) Ff ]k Hp(R) p Littlewood-Paley, ( k )k \local mean kernels" 1 1=2 X ks2 2 kf kHs 2 j k ∗ f (·)j p p k=0 Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 11 / 24 s Fractional smoothness: the Sobolev space Hp Sobolev regularity, 1 < p < 1, s 2 R −1 2 s=2 kf k s := kF [(1 + jξj ) Ff ]k Hp(R) p Littlewood-Paley, ( k )k \local mean kernels" 1 1=2 X ks2 2 kf kHs 2 j k ∗ f (·)j p p k=0 Triebel-Lizorkin spaces, 0 < p; q ≤ 1, s 2 R 1 1=q X ksq q kf kF s := 2 j k ∗ f (·)j p;q p k=0 s The Haar function belongs to Hp if s < 1=p Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 11 / 24 The Haar basis in Sobolev spaces: Sufficiency Theorem (Triebel '10) s The Haar system fhj;k gj2N0;k2Z is an unconditional basis in Hp(R) if max{−1=p0; −1=2g < s < minf1=p; 1=2g : Then 2 1=2 X js2 X j ckf kHs ≤ 2 2 hf ; hj;k iχj;k ≤ Ckf kHs p p p j2N0 k2Z s s 1 1 1 ? 2 + 1 1 p p + 1 1 − 1 2 ? −1 −1 s s Figure: Unconditional basis in Hp Figure: Haar functions in Hp Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 12 / 24 The Haar basis in Sobolev spaces: Necessity Theorem (Seeger, U. '16) s The Haar system fhj;k gj2N0;k2Z is an unconditional basis in Hp(R) if and only if max{−1=p0; −1=2g < s < minf1=p; 1=2g : Then 2 1=2 X js2 X j ckf kHs ≤ 2 2 hf ; hj;k iχj;k ≤ Ckf kHs p p p j2N0 k2Z s s 1 1 1 − 2 + 1 1 p p + 1 1 − 1 2 − −1 −1 s s Figure: Unconditional basis in Hp Figure: Haar functions in Hp Tino Ullrich (Hausdorff-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 13 / 24 The main result: Triebel-Lizorkin spaces Theorem (Seeger, U. '16; Triebel '10) Let 1 < p; q < 1. The Haar system fhj;k gj2N0;k2Z is an unconditional s basis in Fp;q(R) if and only if max{−1=p0; −1=q0g < s < minf1=p; 1=qg : Then the Haar wavelet isomorphism q 1=q X jsq X j ckf kF s ≤ 2 2 hf ; hj;k iχj;k ≤ Ckf kF s p;q p p;q j2N0 k2Z holds true.
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