## Abstract

## Keywords

Martingale inequalities Polynomial spline spaces Orthogonal projection operators## Mathematics Subject Classification

65D07 60G42 42C10## 1 Introduction

*k*be a positive integer, \(({\mathscr {F}}_n)\) an increasing sequence of \(\sigma \)-algebras of sets in [0, 1] where each \({\mathscr {F}}_n\) is generated by a finite partition of [0, 1] into intervals of positive length. Moreover, define the spline space

*k-martingale spline sequence*(adapted to \(({\mathscr {F}}_n)\)) if, for all

*n*,

*k*-martingale spline sequences corresponding to

*arbitrary*filtrations (\({\mathscr {F}}_n\)) of the above type, just by replacing conditional expectation operators by the projection operators \(P_n^{(k)}\). Indeed, we have

- (i)(Shadrin’s theorem) there exists a constant \(C_k\) depending only on
*k*such that$$\begin{aligned} \sup _n\Vert P_n^{(k)} : L_1 \rightarrow L_1 \Vert \le C_k, \end{aligned}$$ - (ii)
(Doob’s weak type inequality for splines)

there exists a constant \(C_k\) depending only on*k*such that for any*k*-martingale spline sequence \((f_n)\) and any \(\lambda >0\),$$\begin{aligned} |\{ \sup _n |f_n| > \lambda \}| \le C_k \frac{\sup _n\Vert f_n\Vert _{1}}{ \lambda }, \end{aligned}$$ - (iii)
(Doob’s \(L_p\) inequality for splines)

for all \(p\in (1,\infty ]\) there exists a constant \(C_{p,k}\) depending only on*p*and*k*such that for all*k*-martingale spline sequences \((f_n)\),$$\begin{aligned} \big \Vert \sup _n |f_n| \big \Vert _{p} \le C_{p,k} \sup _n\Vert f_n\Vert _{p},\ \end{aligned}$$ - (iv)
(Spline convergence theorem)

if \((f_n)\) is an \(L_1\)-bounded

*k*-martingale spline sequence, then \((f_n)\) converges almost surely to some \(L_1\)-function, - (v)
(Spline convergence theorem, \(L_p\)-version)

for \(1<p<\infty \), if \((f_n)\) is an \(L_p\)-bounded

*k*-martingale spline sequence, then \((f_n)\) converges almost surely and in \(L_p\).

*k*-martingale spline sequences with values in a Banach space

*X*characterize the Radon–Nikodým property of

*X*(for background information on that material, we refer to the monographs [6, 20]).

*k*-martingale spline sequences and extend Lépingle’s \(L_1(\ell _2)\)-inequality [12], which reads

*arbitrary*integrable functions \(f_n\):

*p*. This can be seen as a dual version of Doob’s inequality \(\Vert \sup _{\ell } |{\mathbb {E}}[f | {\mathscr {F}}_\ell ]| \Vert _p \le c_p \Vert f\Vert _p\) for \(p>1\), see [1]. Once we know Doob’s inequality for spline projections, which is point (iii) above, the same proof as in [1] works for spline projections if we use suitable positive operators \(T_n\) instead of \(P_n^{(k)}\) that also satisfy Doob’s inequality and dominate the operators \(P_n^{(k)}\) pointwise (cf. Sects. 3.1, 3.2).

## 2 Preliminaries

In this section, we collect all tools that are needed subsequently.

### 2.1 Properties of polynomials

We will need Remez’ inequality for polynomials:

### Theorem 2.1

*p*of order

*k*(i.e. degree \(k-1\)) on

*V*,

Applying this theorem with the set \(E = \{x\in V : |p(x)| \le 8^{-k+1}\Vert p\Vert _{L_\infty (V)} \}\) immediately yields the following corollary:

### Corollary 2.2

*p*be a polynomial of order

*k*on a compact interval \(V\subset {\mathbb {R}}\). Then

### 2.2 Properties of spline functions

For an interval \(\sigma \)-algebra \({\mathscr {F}}\) (i.e. \({\mathscr {F}}\) is generated by a finite collection of intervals having positive length), the space \({\mathscr {S}}_k({\mathscr {F}})\) is spanned by a very special local basis \((N_i)\), the so called B-spline basis. It has the properties that each \(N_i\) is non-negative and each support of \(N_i\) consists of at most *k* neighboring atoms of \({\mathscr {F}}\). Moreover, \((N_i)\) is a partition of unity, i.e. for all \(x\in [0,1]\), there exist at most *k* functions \(N_i\) so that \(N_i(x)\ne 0\) and \(\sum _i N_i(x)=1\). In the following, we denote by \(E_i\) the support of the B-spline function \(N_i\). The usual ordering of the B-splines \((N_i)\)–which we also employ here–is such that for all *i*, \(\inf E_i \le \inf E_{i+1}\) and \(\sup E_i \le \sup E_{i+1}\).

We write \(A(t)\lesssim B(t)\) to denote the existence of a constant *C* such that for all *t*, \(A(t)\le C B(t)\), where *t* denote all implicit and explicit dependencies the expression *A* and *B* might have. If the constant *C* additionally depends on some parameter, we will indicate this in the text. Similarly, the symbols \(\gtrsim \) and \(\simeq \) are used.

Another important property of B-splines is the following relation between B-spline coefficients and the \(L_p\)-norm of the corresponding B-spline expansions.

### Theorem 2.3

*j*,

*k*.

### Theorem 2.4

*f*in the basis \((N_{{\mathscr {F}},i})_i\)

*i*, \(b_i\) is a convex combination of the coefficients \(a_j\) with \({\text {supp}}N_{{\mathscr {G}},j} \supseteq {\text {supp}}N_{{\mathscr {F}},i}\).

For those results and more information on spline functions, in particular B-splines, we refer to [21] or [5].

### 2.3 Spline orthoprojectors

*P*onto \(\mathscr {S}_k({\mathscr {F}})\) in the form

*C*and \(0<q<1\) depending only on

*k*so that for each interval \(\sigma \)-algebra \({\mathscr {F}}\) and each

*i*,

*j*,

### 2.4 Spline square functions

*Sf*is comparable to the \(L_p\)-norm of

*f*:

*p*and

*k*. Moreover, for \(p=1\), it is shown in [9] that

*k*and where the proof of the \(\lesssim \)-part only uses Khintchine’s inequality whereas the proof of the \(\gtrsim \)-part uses fine properties of the functions \(\Delta _n f\).

### 2.5 \(L_p(\ell _q)\)-spaces

## 3 Main results

In this section, we prove our main results. Section 3.1 defines and gives properties of suitable positive operators that dominate our (non-positive) operators \(P_n= P_n^{(k)}\) pointwise. In Sect. 3.2, we use those operators to give a spline version of Stein’s inequality (1.2). A useful property of conditional expectations is the tower property \(\mathbb E_{{\mathscr {G}}} {\mathbb {E}}_{{\mathscr {F}}} f = {\mathbb {E}}_{{\mathscr {G}}} f\) for \({\mathscr {G}}\subset {\mathscr {F}}\). In this form, it extends to the operators \((P_n)\), but not to the operators *T* from Sect. 3.1. In Sect. 3.3 we prove a version of the tower property for those operators. Section 3.4 is devoted to establishing a duality estimate using a spline square function, which is the crucial ingredient in the proofs of the spline versions of both Lépingle’s inequality (1.1) and \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality in Sect. 4.

### 3.1 The positive operators *T*

*q*be a positive number smaller than 1. Then, we define the linear operator \(T = T_{{\mathscr {F}}, q, k}\) by

*T*is selfadjoint (w.r.t the standard inner product on \(L_2\)) and

*T*on \(L_1\) and \(L_\infty \):

*T*is that it is a positive operator, i.e. it maps non-negative functions to non-negative functions and that

*T*satisfies Jensen’s inequality in the form

*x*. This operator is of weak type (1, 1), i.e.

*C*. Since trivially we have the estimate \(\Vert {\mathscr {M}}f\Vert _\infty \le \Vert f\Vert _\infty \), by Marcinkiewicz interpolation, for any \(p>1\), there exists a constant \(C_p\) depending only on

*p*so that

*T*and \({\mathscr {M}}\) at this point is that we can use formula (2.4) and estimate (2.5) to obtain the pointwise bound

*q*given by (2.5), \(C_1\) is a constant that depends only on

*k*and \(C_2\) is a constant that depends only on

*k*and the geometric progression

*q*. But as the parameter \(q<1\) in (2.5) depends only on

*k*, the constant \(C_2\) will also only depend on

*k*.

In other words, (3.3) tells us that the positive operator *T* dominates the non-positive operator *P* pointwise, but at the same time, *T* is dominated by the Hardy–Littlewood maximal function \({\mathscr {M}}\) pointwise and independently of \({\mathscr {F}}\).

### 3.2 Stein’s inequality for splines

We now use this pointwise dominating, positive operator *T* to prove Stein’s inequality for spline projections. For this, let \((\mathscr {F}_n)\) be an interval filtration on [0, 1] and \(P_n\) be the orthogonal projection operator onto the space \({\mathscr {S}}_k(\mathscr {F}_n)\) of splines of order *k* corresponding to \({\mathscr {F}}_n\). Working with the positive operators \(T_{{\mathscr {F}}_n, q, k}\) instead of the non-positive operators \(P_n\), the proof of Stein’s inequality (1.2) for spline projections can be carried over from the martingale case (cf. [1, 24]). For completeness, we include it here.

### Theorem 3.1

*p*,

*r*and

*k*.

### Proof

*q*given by (2.5) instead of the operators \(P_n\). First observe that for \(r=p=1\), the assertion follows from Shadrin’s theorem ((i) on page 1). Inequality (3.3) and the \(L_{p'}\)-boundedness of \({\mathscr {M}}\) for \(1<p'\le \infty \) imply that

*k*. Let \(1\le p<\infty \) and \(U_N : L_{p}(\ell _1^N) \rightarrow L_{p}\) be given by \((g_1,\ldots ,g_N)\mapsto \sum _{j=1}^N T_j g_j\). Inequality (3.5) implies the boundedness of the adjoint \(U_N^* : L_{p'}\rightarrow L_{p'}(\ell _\infty ^N)\), \(f\mapsto (T_j f)_{j=1}^N\) for \(p'=p/(p-1)\) by a constant independent of

*N*and therefore also the boundedness of \(U_N\). Since \(|T_j f| \le T_j|f|\) by the positivity of \(T_j\), letting \(N\rightarrow \infty \) implies (3.4) for \(T_n\) instead of \(P_n\) in the case \(r=1\) and outer parameter \(1\le p < \infty \).

*p*/

*r*to get the result for \(1\le r\le p<\infty \). The cases \(1<p\le r\le \infty \) now just follow from this result using duality and the self-adjointness of \(T_j\). \(\square \)

### 3.3 Tower property of *T*

Next, we will prove a substitute of the tower property \(\mathbb E_{{\mathscr {G}}} {\mathbb {E}}_{{\mathscr {F}}} f={\mathbb {E}}_{{\mathscr {G}}}f\)\(({\mathscr {G}}\subset {\mathscr {F}})\) for conditional expectations that applies to the operators *T*.

*k*

*-regularity parameter*\(\gamma _k({\mathscr {F}})\) is defined as

*i*so that \(E_i\) and \(E_{i+1}\) are defined. The name

*k*-regularity is motivated by the fact that each B-spline support \(E_i\) of order

*k*consists of at most

*k*(neighboring) atoms of the \(\sigma \)-algebra \({\mathscr {F}}\).

### Proposition 3.2

*T*) Let \({\mathscr {G}}\subset {\mathscr {F}}\) be two interval \(\sigma \)-algebras on [0, 1]. Let \(S = T_{{\mathscr {G}},\sigma ,k}\) and \(T=T_{\mathscr {F},\tau ,k'}\) for some \(\sigma ,\tau \in (0,1)\) and some positive integers \(k,k'\). Then, for all \(q>\max (\tau ,\sigma )\), there exists a constant

*C*depending on \(q,k,k'\) so that

*k*-regularity parameter of \({\mathscr {G}}\).

### Proof

Let \((F_i)\) be the collection of B-spline supports in \(\mathscr {S}_{k'}({\mathscr {F}})\) and \((G_i)\) the collection of B-spline supports in \({\mathscr {S}}_{k}({\mathscr {G}})\). Moreover, we denote by \(F_{ij}\) the smallest interval containing \(F_i\) and \(F_j\) and by \(G_{ij}\) the smallest interval containing \(G_i\) and \(G_j\).

*S*and \(K_T\) of

*T*(cf. 3.1)

*C*depending on \(q,k,k'\). In order to prove this inequality, we first fix \(x,s\in [0,1]\) and choose

*i*such that \(x\in G_i\) and \(\ell \) such that \(s\in F_\ell \). Moreover, based on \(\ell \), we choose

*j*so that \(s\in G_j\) and \(G_j \supset F_\ell \). There are at most \(\max (k,k')\) choices for each of the indices \(i,\ell ,j\) and without restriction, we treat those choices separately, i.e. we only have to estimate the expression

*r*, there are also at most \(k+k'-1\) indices

*m*so that \(|G_m\cap F_r| >0\) (recall that \({\mathscr {G}}\subset \mathscr {F}\)), we choose one such index \(m=m(r)\) and estimate

*r*in the above sum,

*k*-regularity parameter \(\gamma = \gamma _{k}({\mathscr {G}})\) of the \(\sigma \)-algebra \({\mathscr {G}}\) and denoting \(\lambda = \max (\tau ,\sigma )\), we estimate by

*r*as in the sum) than atoms of \({\mathscr {G}}\) between \(G_m\) and \(G_j\). Finally, we see that for any \(q>\lambda \),

*C*depending on \(q,k,k'\), and, as \(x\in G_i\) and \(s\in G_j\), this shows inequality (3.7). \(\square \)

As a corollary of Proposition 3.2, we have

### Corollary 3.3

Let \((f_n)\) be functions in \(L_1\). We denote by \(P_n\) the orthogonal projection onto \({\mathscr {S}}_{k}({\mathscr {F}}_n)\) and by \(P_n'\) the orthogonal projection onto \({\mathscr {S}}_{k'}({\mathscr {F}}_n)\) for some positive integers \(k,k'\). Moreover, let \(T_n\) be the operator \(T_{{\mathscr {F}}_n, q, k}\) from (3.3) dominating \(P_n\) pointwise.

*n*and for any \(1\le p\le \infty \),

*k*and \(k'\).

We remark that by Jensen’s inequality and the tower property, this is trivial for conditional expectations \({\mathbb {E}}(\cdot | \mathscr {F}_n)\) instead of the operators \(P_n, T_n, P_{\ell -1}'\) even with an absolute constant on the right hand side.

### Proof

*k*and \(k'\) respectively. Setting \(U_n := T_{{\mathscr {F}}_n, \max (q,q')^{1/2}, k}\), we perform the following chain of inequalities, where we use the positivity of \(T_n\) and (3.3), Jensen’s inequality for \(T_{\ell -1}'\), the tower property for \(T_n T_{\ell -1}'\) and the \(L_p\)-boundedness of \(U_n\), respectively:

*k*and \(k'\). \(\square \)

### 3.4 A duality estimate using a spline square function

In order to give the desired duality estimate contained in Theorem 3.6, we need the following construction of a function \(g_n\in {\mathscr {S}}_k({\mathscr {F}}_n)\) based on a spline square function.

### Proposition 3.4

*n*and set

*n*,

- (1)
\(g_n \le g_{n+1}\),

- (2)
\(X_n^{1/2} \le g_n\)

- (3)
\({\mathbb {E}} g_n \lesssim {\mathbb {E}} X_n^{1/2}\), where the implied constant depends on

*k*and on \(\sup _{m\le n}\gamma _{k}({\mathscr {F}}_m)\).

For the proof of this result, we need the following simple lemma.

### Lemma 3.5

- (1)
\(|\varphi (j)| = c_1 |A_j|\) for all

*j*, - (2)
\(\varphi (j) \subseteq B_j\) for all

*j*, - (3)
\(\varphi (i) \cap \varphi (j) = \emptyset \) for all \(i\ne j\).

### Proof

- (1)
\(|\varphi (j)| = c_1|A_j|\),

- (2)
\(\varphi (j)\subseteq B_j\),

- (3)
\(\varphi (j) \cap \cup _{i<j} \varphi (i) = \emptyset \).

### Proof of Proposition 3.4

*n*and let \((N_{n,j})\) be the B-spline basis of \(\mathscr {S}_{k}({\mathscr {F}}_n)\). Moreover, for any

*j*, set \(E_{n,j} = {\text {supp}}N_{n,j}\) and \(a_{n,j} := \max _{\ell \le n} \max _{r : E_{\ell ,r} \supset E_{n,j}}\Vert X_{\ell } \Vert _{L_\infty (E_{\ell ,r})}^{1/2}\) and we define \(\ell (j)\le n\) and

*r*(

*j*) so that \(E_{\ell (j),r(j)} \supseteq E_{n,j} \) and \(a_{n,j} = \Vert X_{\ell (j)} \Vert _{L_\infty (E_{\ell (j),r(j)})}^{1/2}\). Set

*k*and on \(\sup _{m\le n} \gamma _k({\mathscr {F}}_m)\). By B-spline stability (Theorem 2.3), we estimate the integral of \(g_n\) by

*k*. In order to continue the estimate, we next show the inequality

*k*. Indeed, we use Theorem 2.3 in the form of (2.3) to get (\(f_m\in {\mathscr {S}}_k({\mathscr {F}}_\ell )\) for \(m\le \ell \))

*I*of \({\mathscr {F}}_\ell \), due to the equivalence of

*p*-norms of polynomials (cf. Corollary 2.2),

*k*indices

*s*so that \(|E_{\ell ,s} \cap E_{\ell ,r}|>0\), we have established (3.10).

*j*and let

*i*be such that \(\ell (i)\ge \ell (j)\). By definition of \(D_i = K_{\ell (i), r(i)}\), the smallest interval containing \(J_{n,i}\) and \(D_i\) contains at most \(2k-1\) atoms of \({\mathscr {F}}_{\ell (i)}\) and, if \(D_{i}\subset D_j\), the smallest interval containing \(J_{n,i}\) and \(D_j\) contains at most \(2k-1\) atoms of \({\mathscr {F}}_{\ell (j)}\). This means that, in particular, \(J_{n,i}\) is a subset of the union

*V*of 4

*k*atoms of \({\mathscr {F}}_{\ell (j)}\) with \(D_j\subset V\). Since each atom of \({\mathscr {F}}_n\) occurs at most

*k*times in the sequence \((A_j)\), there exists a constant \(c_1\) depending on

*k*and \(\sup _{u\le \ell (j)} \gamma _k({\mathscr {F}}_u)\le \sup _{u\le n} \gamma _k(\mathscr {F}_u)\) so that

*i*,

*j*. Using these properties of \(\varphi \), we continue the estimate in (3.12) and write

*k*and \(\sup _{u\le n}\gamma _k({\mathscr {F}}_u)\). \(\square \)

Employing this construction of \(g_n\), we now give the following duality estimate for spline projections (for the martingale case, see for instance [8]). The martingale version of this result is the essential estimate in the proof of both Lépingle’s inequality (1.1) and the \(H^1\)-\({{\,\mathrm{BMO}\,}}\) duality.

### Theorem 3.6

*n*. Additionally, let \(h_n\in L_1\) be arbitrary. Then, for any

*N*,

*k*and \(\gamma \).

### Proof

*k*and \(\gamma \). Then, apply Cauchy–Schwarz inequality by introducing the factor \(g_n^{1/2}\) to get

## 4 Applications

We give two applications of Theorem 3.6, (i) D. Lépingle’s inequality and (ii) an analogue of C. Fefferman’s \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality in the setting of splines. Once the results from Sect. 3 are known, the proofs of the subsequent results proceed similarly to their martingale counterparts in [8, 12] by using spline properties instead of martingale properties.

### 4.1 Lépingle’s inequality for splines

### Theorem 4.1

*n*, \(f_n \in {\mathscr {S}}_{k}({\mathscr {F}}_n)\) and \(P_n'\) be the orthogonal projection operator on \({\mathscr {S}}_{k'}({\mathscr {F}}_n)\). Then,

*k*, \(k'\) and \(\sup _n \gamma _{k}({\mathscr {F}}_n)\).

We emphasize that the parameters *k* and \(k'\) can be different here, *k* being the spline order of the sequence \((f_n)\) and \(k'\) being the spline order of the projection operators \(P_{n-1}'\). In particular, the constant on the right hand side does not depend on the \(k'\)-regularity parameter \(\sup _n \gamma _{k'}({\mathcal {F}}_n)\).

### Proof

*n*, by Corollary 3.3,

*k*,\(k'\) and \(\sup _{n\le N} \gamma _k({\mathscr {F}}_n)\). Letting

*N*tend to infinity, we obtain the conclusion. \(\square \)

### 4.2 \(H_1\)-\({{\,\mathrm{BMO}\,}}\) duality for splines

*k*and the orthogonal projection operators \(P_n\) onto \({\mathscr {S}}_k({\mathscr {F}}_n)\) and additionally, we set \(P_0=0\). For \(f\in L_1\), we introduce the notation

*V*be the \(L_1\)-closure of \(\cup _n {\mathscr {S}}_k({\mathscr {F}}_n)\). Then, the uniform boundedness of \(P_n\) on \(L_1\) implies that \(P_n f\rightarrow f\) in \(L_1\) for \(f\in V\). Next, set

Let us now assume \(\sup _n \gamma _k({\mathscr {F}}_n) < \infty \). In this case we identify, similarly to \(H_1\)-\({{\,\mathrm{BMO}\,}}\)-duality (cf. [7, 8, 10]), \({{\,\mathrm{BMO}\,}}_k\) as the dual space of \(H_{1,k}\).

*L*on \(H_{1,k}\), we extend

*L*norm-preservingly to a continuous linear functional \(\Lambda \) on \(L_1(\ell _2)\), which, by Sect. 2.5, has the form

*k*-martingale spline sequence \(h_n= \sum _{\ell \le n} \Delta _\ell \sigma _\ell \) is bounded in \(L_2\) and therefore, by the spline convergence theorem ((v) on page 2), has a limit \(h\in L_2\) with \(P_n h = h_n\) and which is also contained in \({{\,\mathrm{BMO}\,}}_k\). Indeed, by using Corollary 3.3, we obtain \(\Vert h\Vert _{{{\,\mathrm{BMO}\,}}_k} \lesssim \Vert \sigma \Vert _{L_\infty (\ell _2)} = \Vert \Lambda \Vert = \Vert L\Vert \) with a constant depending only on

*k*and \(\sup _{n}\gamma _k({\mathscr {F}}_n)\). Moreover, for \(f\in H_{1,k}\), since

*L*is continuous on \(H_{1,k}\),

### Theorem 4.2

*k*and \(\sup _n \gamma _k({\mathscr {F}}_n)\).

### Remark 4.3

- (1)By Khintchine’s inequality, \(\Vert Sf\Vert _1 \lesssim \sup _{\varepsilon \in \{-1,1\}^{{\mathbb {Z}}}} \Vert \sum _n\varepsilon _n\Delta _n f\Vert _1\). Based on the interval filtration \(({\mathscr {F}}_n)\), we can generate an interval filtration \(({\mathscr {G}}_n)\) that contains \(({\mathscr {F}}_n)\) as a subsequence and each \({\mathscr {G}}_{n+1}\) is generated from \({\mathscr {G}}_n\) by dividing exactly one atom of \({\mathscr {G}}_n\) into two atoms of \(\mathscr {G}_{n+1}\). Denoting by \(P_n^{{\mathscr {G}}}\) the orthogonal projection operator onto \({\mathscr {S}}_k({\mathscr {G}}_n)\) and \(\Delta _j^{\mathscr {G}}= P_j^{{\mathscr {G}}}-P_{j-1}^{{\mathscr {G}}}\), we can writefor some sequence \((a_n)\). By using inequalities (2.7) and (2.6) and writing \((S^{{\mathscr {G}}}f)^2= \sum _j |\Delta _j^{{\mathscr {G}}} f|^2\), we obtain for \(p>1\)$$\begin{aligned} \sum _n\varepsilon _n\Delta _n f = \sum _n \varepsilon _n \sum _{j=a_n}^{a_{n+1}-1} \Delta _j^{{\mathscr {G}}} f \end{aligned}$$This implies \(L_p\subset H_{1,k}\) for all \(p>1\) and, by duality, \({{\,\mathrm{BMO}\,}}_k \subset L_p\) for all \(p<\infty \).$$\begin{aligned} \Vert Sf\Vert _1 \lesssim \Vert S^{{\mathscr {G}}} f\Vert _1 \le \Vert S^{{\mathscr {G}}} f\Vert _p \lesssim \Vert f\Vert _p. \end{aligned}$$
- (2)If \(({\mathscr {F}}_n)\) is of the form that each \(\mathscr {F}_{n+1}\) is generated from \({\mathscr {F}}_n\) by splitting exactly one atom of \({\mathscr {F}}_n\) into two atoms of \({\mathscr {F}}_{n+1}\) and under the condition \(\sup _n \gamma _{k-1}({\mathscr {F}}_n) < \infty \) (which is stronger than \(\sup _n \gamma _k({\mathscr {F}}_n)<\infty \)), it is shown in [9] thatwhere \(H_1\) denotes the atomic Hardy space on [0, 1], i.e. in this case, \(H_{1,k}\) coincides with \(H_1\).$$\begin{aligned} \Vert Sf\Vert _{1} \simeq \Vert f\Vert _{H_1}, \end{aligned}$$

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). It is a pleasure to thank P. F. X. Müller for very helpful conversations during the preparation of this paper. The author is supported by the Austrian Science Fund (FWF), Project F5513-N26, which is a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.

## References

- 1.Asmar, N., Montgomery-Smith, S.: Littlewood-Paley theory on solenoids. Colloq. Math.
**65**(1), 69–82 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 2.Benedek, A., Panzone, R.: The space \(L^{p}\), with mixed norm. Duke Math. J.
**28**, 301–324 (1961)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Bourgain, J.: Embedding \(L^{1}\) in \(L^{1}/H^{1}\). Trans. Am. Math. Soc.
**278**(2), 689–702 (1983)MathSciNetGoogle Scholar - 4.Delbaen, F., Schachermayer, W.: An inequality for the predictable projection of an adapted process. In: Séminaire de Probabilités, XXIX, volume 1613 of Lecture Notes in Mathematics, pp. 17–24. Springer, Berlin (1995)Google Scholar
- 5.De Vore, R.A., Lorentz, G.G.: Constructive approximation, volume 303 Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1993)Google Scholar
- 6.Diestel, J., Uhl Jr., J.J.: Vector Measures. American Mathematical Society, Providence (1977). (
**With a foreword by B. J. Pettis, Mathematical Surveys, No. 15**)CrossRefzbMATHGoogle Scholar - 7.Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc.
**77**, 587–588 (1971)MathSciNetCrossRefzbMATHGoogle Scholar - 8.Garsia, A.M.: Martingale Inequalities: Seminar Notes on Recent Progress. Mathematics Lecture Notes Series. W. A. Benjamin, Inc., Reading (1973)zbMATHGoogle Scholar
- 9.Gevorkyan, G., Kamont, A., Keryan, K., Passenbrunner, M.: Unconditionality of orthogonal spline systems in \(H^1\). Stud. Math.
**226**(2), 123–154 (2015)CrossRefzbMATHGoogle Scholar - 10.Herz, C.: Bounded mean oscillation and regulated martingales. Trans. Am. Math. Soc.
**193**, 199–215 (1974)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Keryan, K., Passenbrunner, M.: Unconditionality of Periodic Orthonormal Spline Systems in \({L}^p\) (2017). arXiv:1708.09294 (to appear in Studia Mathematica)
- 12.Lépingle, D.: Une inégalité de martingales. In:
*Séminaire de Probabilités, XII (University of Strasbourg, Strasbourg, 1976/1977)*, Volume 649 of Lecture Notes in Mathematics, pp. 134–137. Springer, Berlin (1978)Google Scholar - 13.Müller, P.F.X.: A decomposition for Hardy martingales. Indiana Univ. Math. J.
**61**(5), 1801–1816 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Müller, P.F.X., Passenbrunner, M.: Almost Everywhere Convergence of Spline Sequences (2017). arXiv:1711.01859
- 15.Neveu, J.: Discrete-Parameter Martingales, revised edn. North-Holland Publishing Co., Amsterdam (1975). (
**Translated from the French by T. P. Speed, North-Holland Mathematical Library, Vol. 10**)zbMATHGoogle Scholar - 16.Passenbrunner, M.: Unconditionality of orthogonal spline systems in \(L^p\). Stud. Math.
**222**(1), 51–86 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Passenbrunner, M.: Orthogonal projectors onto spaces of periodic splines. J. Complex.
**42**, 85–93 (2017)MathSciNetCrossRefzbMATHGoogle Scholar - 18.Passenbrunner, M.: Spline Characterizations of the Radon–Nikodým Property (2018). arXiv:1807.01861
- 19.Passenbrunner, M., Shadrin, A.: On almost everywhere convergence of orthogonal spline projections with arbitrary knots. J. Approx. Theory
**180**, 77–89 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Pisier, G.: Martingales in Banach Spaces. Volume 155 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2016)Google Scholar
- 21.Schumaker, L.L.: Spline Functions: Basic Theory. Pure and Applied Mathematics. Wiley, New York (1981)Google Scholar
- 22.Shadrin, A.: The \(L_\infty \)-norm of the \(L_2\)-spline projector is bounded independently of the knot sequence: a proof of de Boor’s conjecture. Acta Math.
**187**(1), 59–137 (2001)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30. Princeton University Press, Princeton (1970)Google Scholar
- 24.Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton (1970)Google Scholar
- 25.von Golitschek, M.: On the \(L_\infty \)-norm of the orthogonal projector onto splines. A short proof of A. Shadrin’s theorem. J. Approx. Theory
**181**, 30–42 (2014)Google Scholar

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