Haar Projection Numbers and Failure of Unconditional Convergence

Haar projection numbers and failure of unconditional convergence

Tino Ullrich

Hausdorff-Zentrum fürMathematik / Institut fürNumerische Simulation UniversitätBonn

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)

x 1 1 2

−1

The univariate Haar wavelet system

j hj,k (x) := h(2 x − k) , j ∈ N0, k ∈ Z .

j/2 The system {2 hj,k }j,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 ∈ N0, k ∈ 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 ∈ N0, k ∈ 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 ∈ C(T)

∞ 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 ∈ C(T)

∞ 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 Point clouds

Digital net: Sparse grid: Optimal cubature Optimal recovery of functions

Digital net, d = 2, N = 256 Sparse grid, d = 2, N = 256 1 1

3 3 4 4

1 1 2 2

1 1 4 4

0 0 0 1 1 3 1 0 1 1 3 1 4 2 4 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 < ∞

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 < ∞

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 {e }k∈Z the Haar wavelet system is an unconditional basis in Lp for p 6= 2

Theorem (Marcinkiewicz ’37, Paley ’32) Let 1 < p < ∞.

(i) The Haar system {hj,k }j,k∈Z 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 j∈Z k∈Z

The statement does neither hold in case p = 1 nor p = ∞ !

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 10 / 24 Littlewood-Paley, (ψk )k “local mean kernels”

∞ 1/2 X ks2 2 kf kHs 2 |ψk ∗ f (·)| p p k=0

Triebel-Lizorkin spaces, 0 < p, q ≤ ∞, s ∈ R ∞ 1/q X ksq q kf kF s := 2 |ψk ∗ f (·)| 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 < ∞, s ∈ R −1 2 s/2 kf k s := kF [(1 + |ξ| ) 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 ≤ ∞, s ∈ R ∞ 1/q X ksq q kf kF s := 2 |ψk ∗ f (·)| 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 < ∞, s ∈ R −1 2 s/2 kf k s := kF [(1 + |ξ| ) Ff ]k Hp(R) p

Littlewood-Paley, (ψk )k “local mean kernels”

∞ 1/2 X ks2 2 kf kHs 2 |ψk ∗ f (·)| 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 < ∞, s ∈ R −1 2 s/2 kf k s := kF [(1 + |ξ| ) Ff ]k Hp(R) p

Littlewood-Paley, (ψk )k “local mean kernels”

∞ 1/2 X ks2 2 kf kHs 2 |ψk ∗ f (·)| p p k=0

Triebel-Lizorkin spaces, 0 < p, q ≤ ∞, s ∈ R ∞ 1/q X ksq q kf kF s := 2 |ψk ∗ f (·)| 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 {hj,k }j∈N0,k∈Z is an unconditional basis in Hp(R) if max{−1/p0, −1/2} < s < min{1/p, 1/2} . Then

2 1/2 X js2 X j ckf kHs ≤ 2 2 hf , hj,k iχj,k ≤ Ckf kHs p p p j∈N0 k∈Z 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 {hj,k }j∈N0,k∈Z is an unconditional basis in Hp(R) if and only if max{−1/p0, −1/2} < s < min{1/p, 1/2} . Then

2 1/2 X js2 X j ckf kHs ≤ 2 2 hf , hj,k iχj,k ≤ Ckf kHs p p p j∈N0 k∈Z 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 < ∞. The Haar system {hj,k }j∈N0,k∈Z is an unconditional s basis in Fp,q(R) if and only if

max{−1/p0, −1/q0} < s < min{1/p, 1/q} .

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 j∈N0 k∈Z holds true.

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 14 / 24 Wavelet isomorphism ?

s Observation: non-equivalence of the norms k · kHp and

q 1/q X jsq X j k · k s,dyad := 2 2 hf , hj,k iχj,k Fp,q p j∈N0 k∈Z in the lower triangle (question marked region) =⇒ Wavelet isomorphism fails!

Theorem (Seeger, U. 2016) Let q ≤ p < ∞ and −1 < s ≤ −1/q0. Then there exists a sequence of test functions {FN }N such that (i) kF k s ≤ C(s, p, q) , N Fp,q N(−1/q0−s) 1/q (ii) kFN k s,dyad max{2 , N } . Fp,q &

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 15 / 24 Unconditional bases

Deﬁnition ∗ Let X be a Banach space. Any biorthogonal system (xn, xn ) with a countable index set A is an unconditional basis in X if

span{xn}n = X There exists a constant C > 0 such that for every x ∈ X and every ﬁnite index set B ⊂ A

X ∗ xn (x)xn ≤ CkxkX X n∈B

In other words: the norm of any projection

X ∗ PB (x) := xn (x)xn n∈B is uniformly bounded by C

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 16 / 24 Large projection norms

Let E ⊂ {hj,k }j∈N−1,k∈Z a ﬁnite subset of the Haar system Associated projections

X j PE (f ) := 2 hf , hj,k ihj,k

hj,k ∈E “grow” with #E (in the “lower” und “upper” triangle) s 1 1 − 2 + 1 Probabilistic construction p + 1 Quantitative version − 1 2 − (sharp growing rates) −1

s Figure: Unconditional basis in Hp

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 17 / 24 A probabilistic argument

#A ∼ 2N+1 Random operator

2j −1 X X j TA,εg(x) = εj 2 hhj,µ, gihj,µ(x) 2j ∈A µ=0 Random test function

2k X 0 −(k+N)r X fA,ε0 = εk 2 ηk+N,2N µ 2k ∈A µ=1 Via Khinchine’s inequality

1/2 2 N(−s−1/2) ε0 ε TA,εfA,ε0 s ≥ c2 E E Hp

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 18 / 24 Haar projection numbers γ∗(X ; Λ) = sup G(X , A):#A ≤ Λ} , γ∗(X ; Λ) = inf G(X , A):#A ≥ Λ} .

Theorem (Seeger, U. 2015) (i) For 1 < p < q < ∞, 1/q < s < 1/p,

1 s ∗ s s− q γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

(ii) For 1 < q < p < ∞, −1/p0 < s < −1/q0,

− 1 −s s ∗ s q0 γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

∗ Endpoint case: γ∗ and γ diﬀer, but grow in log Λ

Haar projection numbers: Triebel-Lizorkin spaces

G(X , A) = sup kPE kX →X : HF(E) ⊂ A

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 19 / 24 Theorem (Seeger, U. 2015) (i) For 1 < p < q < ∞, 1/q < s < 1/p,

1 s ∗ s s− q γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

(ii) For 1 < q < p < ∞, −1/p0 < s < −1/q0,

− 1 −s s ∗ s q0 γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

∗ Endpoint case: γ∗ and γ diﬀer, but grow in log Λ

Haar projection numbers: Triebel-Lizorkin spaces

G(X , A) = sup kPE kX →X : HF(E) ⊂ A Haar projection numbers γ∗(X ; Λ) = sup G(X , A):#A ≤ Λ} , γ∗(X ; Λ) = inf G(X , A):#A ≥ Λ} .

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 19 / 24 ∗ Endpoint case: γ∗ and γ diﬀer, but grow in log Λ

Haar projection numbers: Triebel-Lizorkin spaces

G(X , A) = sup kPE kX →X : HF(E) ⊂ A Haar projection numbers γ∗(X ; Λ) = sup G(X , A):#A ≤ Λ} , γ∗(X ; Λ) = inf G(X , A):#A ≥ Λ} .

Theorem (Seeger, U. 2015) (i) For 1 < p < q < ∞, 1/q < s < 1/p,

1 s ∗ s s− q γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

(ii) For 1 < q < p < ∞, −1/p0 < s < −1/q0,

− 1 −s s ∗ s q0 γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 19 / 24 Haar projection numbers: Triebel-Lizorkin spaces

G(X , A) = sup kPE kX →X : HF(E) ⊂ A Haar projection numbers γ∗(X ; Λ) = sup G(X , A):#A ≤ Λ} , γ∗(X ; Λ) = inf G(X , A):#A ≥ Λ} .

Theorem (Seeger, U. 2015) (i) For 1 < p < q < ∞, 1/q < s < 1/p,

1 s ∗ s s− q γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

(ii) For 1 < q < p < ∞, −1/p0 < s < −1/q0,

− 1 −s s ∗ s q0 γ∗(Fp,q; Λ) ≈ γ (Fp,q; Λ) ≈ Λ .

∗ Endpoint case: γ∗ and γ diﬀer, but grow in log Λ

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 19 / 24 Deterministic examples of large Haar projections

N #AN 2 0 0 Lacunary in Haar frequency: if `, ` ∈ AN then |j − j | > R Haar projection

2j −1 X X j PAN f := 2 hf , hj,mihj,m j∈AN m=0 Test functions

2` X −(`+N)s X fN (x) := 2 η`+N,2N µ+2N−1 `∈AN µ=0 satisfy a uniform bound independently of N if p > q

kf k s ≤ C(s, p, q) N Fp,q (Frazier-Jawerth atoms + Feﬀerman-Stein interpolation theorem)

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 20 / 24 Recall 2` X −(`+N)s X fN (x) := 2 η`+N,2N µ `∈AN µ=0

Idea of proof I

We want to have ∞ 1/q 0 X ksq q N(−1/q −s) 2 |ψk ∗ PAN fN | & 2 p k=0 It suﬃces to prove (recall p > q)

1/q 0 X ksq q N(−1/q −s) 2 |ψk ∗ PAN fN | & 2 q k∈AN

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 21 / 24 Idea of proof I

We want to have ∞ 1/q 0 X ksq q N(−1/q −s) 2 |ψk ∗ PAN fN | & 2 p k=0 It suﬃces to prove (recall p > q)

1/q 0 X ksq q N(−1/q −s) 2 |ψk ∗ PAN fN | & 2 q k∈AN

Recall 2` X −(`+N)s X fN (x) := 2 η`+N,2N µ `∈AN µ=0

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 21 / 24 and the “oﬀ diagonal terms”

q 1/q X ksq X −`s j 2 2 h2 η`,µ, hj,mihj,m ∗ ψk q k∈AN (j,m),(`,µ) (j,`)6=(k,k+N)

from above by2 −Rε2N(−1/q0−s) N(−1/q0−s) Alltogether & CR 2 if R > 0 is chosen large enough!

Idea of proof II

Using various cancellation properties we bound the “diagonal terms”

q 1/q X ksq X X −(k+N)s 2 h2 ηk+N,µ, hk,mihk,m ∗ ψk q k∈AN m µ

from below by R−1/q2N(−1/q0−s)

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 22 / 24 N(−1/q0−s) Alltogether & CR 2 if R > 0 is chosen large enough!

Idea of proof II

Using various cancellation properties we bound the “diagonal terms”

q 1/q X ksq X X −(k+N)s 2 h2 ηk+N,µ, hk,mihk,m ∗ ψk q k∈AN m µ

from below by R−1/q2N(−1/q0−s) and the “oﬀ diagonal terms”

q 1/q X ksq X −`s j 2 2 h2 η`,µ, hj,mihj,m ∗ ψk q k∈AN (j,m),(`,µ) (j,`)6=(k,k+N)

from above by2 −Rε2N(−1/q0−s)

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 22 / 24 Idea of proof II

Using various cancellation properties we bound the “diagonal terms”

q 1/q X ksq X X −(k+N)s 2 h2 ηk+N,µ, hk,mihk,m ∗ ψk q k∈AN m µ

from below by R−1/q2N(−1/q0−s) and the “oﬀ diagonal terms”

q 1/q X ksq X −`s j 2 2 h2 η`,µ, hj,mihj,m ∗ ψk q k∈AN (j,m),(`,µ) (j,`)6=(k,k+N)

from above by2 −Rε2N(−1/q0−s) N(−1/q0−s) Alltogether & CR 2 if R > 0 is chosen large enough!

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 22 / 24 Theorem (Garrig´os,Seeger, U.) Let 1 < p, q < ∞ and −1/p0 < s < 1p. Then there is an enumeration {u1, u2, u3, ...} of the Haar system {hj,k }j,k such that

kf − P f k s → 0 if N → ∞ , N Fp,q

N P −2 where PN f := kunk hf , uniun n=1

Admissible enumeration U = {u1, u2, ...} of the Haar system: 0 j < j , un = hj,k , un0 = hj0,k0 ,

supp(un) ⊂ [ν, ν + 1), supp(un0 ) ⊂ [ν, ν + 1), then n ≤ n0

Schauder basis

We have seen: for certain parameters the Haar system is not an s s unconditional basis in Fp,q (and even Hp)

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 23 / 24 Admissible enumeration U = {u1, u2, ...} of the Haar system: 0 j < j , un = hj,k , un0 = hj0,k0 ,

supp(un) ⊂ [ν, ν + 1), supp(un0 ) ⊂ [ν, ν + 1), then n ≤ n0

Schauder basis

We have seen: for certain parameters the Haar system is not an s s unconditional basis in Fp,q (and even Hp) Theorem (Garrig´os,Seeger, U.) Let 1 < p, q < ∞ and −1/p0 < s < 1p. Then there is an enumeration {u1, u2, u3, ...} of the Haar system {hj,k }j,k such that

kf − P f k s → 0 if N → ∞ , N Fp,q

N P −2 where PN f := kunk hf , uniun n=1

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 23 / 24 Schauder basis

kf − P f k s → 0 if N → ∞ , N Fp,q

N P −2 where PN f := kunk hf , uniun n=1

Admissible enumeration U = {u1, u2, ...} of the Haar system: 0 j < j , un = hj,k , un0 = hj0,k0 ,

supp(un) ⊂ [ν, ν + 1), supp(un0 ) ⊂ [ν, ν + 1), then n ≤ n0

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 23 / 24 Thank you for your attention!

Tino Ullrich (Hausdorﬀ-Zentrum Uni Bonn) Haar projection numbers Strobl, June 2016 24 / 24