# Multivariate Haar systems in Besov function spaces

We determine all cases for which the d-dimensional Haar wavelet system H^d on the unit cube I^d is a conditional or unconditional Schauder basis in the classical isotropic Besov function spaces B_p,q,1^s(I^d), 0<p,q<∞, 0< s < 1/p, defined in terms of first-order L_p moduli of smoothness. We obtain similar results for the tensor-product Haar system H̃^d, and characterize the parameter range for which the dual of B_p,q,1^s(I^d) is trivial for 0<p<1.

## Authors

• 1 publication
07/08/2019

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## 1 Introduction

The univariate Haar system was one of the first examples of a Schauder basis in some classical function spaces on the unit interval , see Ci1977 , (KaSa1989, , Section III), and (Tr2010, , Section 2.1) for a review of the early history of the Haar system as basis in function spaces. Meantime the existence of Schauder bases in function spaces of Besov-Hardy-Sobolev type has been established in most cases, see Tr2006 for a recent exposition. Early on, a major step was taken by Ciesielski and co-workers CiDo1972 ; Ci1975 ; Ci1977 ; CiFi1982 ; CiFi1983a ; CiFi1983b who constructed families of spline systems generalizing the classical Haar, Faber, and Franklin systems, and established their basis properties in Lebesgue-Sobolev spaces over -dimensional cubes and smooth manifolds for . For distributional Besov spaces and Triebel-Lizorkin spaces with , , wavelet systems provide examples of unconditional Schauder bases. Results in these directions and their generalization to spaces on domains in are presented in Tr2008 ; Tr2010 . Needless to say that not all quasi-Banach function spaces possess nice basis properties. E.g., does not possess an unconditional Schauder basis, see (KaSa1989, , Theorem II.13), while the quasi-Banach space , , cannot have Schauder bases at all since its dual is trivial.

In this paper, we deal with the multivariate anisotropic tensor-product Haar system

 ~Hd=H⊗…⊗Hd times,d≥1, (1)

and its isotropic counterpart on the unit cube (the latter is called Haar wavelet system in Tr2010 ), and consider their Schauder basis properties in the Besov spaces . These function spaces are classically defined in terms of first-order moduli of smoothness (detailed definitions are given in the next section), and coincide with their distributional counterparts only under some restrictions on . We mostly concentrate on the parameter range

 0

With the exception of the special case , this is the maximal range of parameters for which is a separable quasi-Banach space and contains the Haar systems and . Moreover, for this parameter range admits a characterization in terms of best -approximations with piecewise constant functions on dyadic partitions of which is used in the proofs. Our main result for the Haar wavelet system is the following theorem.

###### Theorem 1

Assume (2).
a) If then the Haar wavelet system is an unconditional Schauder basis for for all parameters , of interest.
b) Let . Then is an unconditional Schauder basis for if and only if , . If , then is a Schauder basis for but not unconditional. In all other cases, does not possess the Schauder basis property for .

The statements about unconditionality of in Theorem 1 are proved by giving a characterization of in terms of Haar coefficients. We refer to Theorem 3 in Section 3.2 which states conditions under which is isomorphic to some weighted sequence space. The exceptional case is covered by Theorem 4 in Section 5.2. We also have

###### Theorem 2

If in (2) then the Besov space does not possess nontrivial bounded linear functionals, i.e., , if and only if , or , .

Consequently, for these parameters does not have a Schauder basis at all which is a stronger statement than proving that the Haar wavelet system fails to be a Schauder basis. For the parameters

 0

where according to Theorem 1 is not a Schauder basis in , we have the continuous embedding , and thus . We do not know if the spaces satisfying (3) possess (unconditional) Schauder bases at all.

As Schauder basis in Besov spaces , the tensor-product Haar system behaves the same as for but fails completely for the parameter arnge . This is the essence of Theorem 5 in Section 5.3.

Let us comment on known results that motivated this study. For , the two Haar systems and have formally been introduced in the 1970-ies, see Ci1977 ; CiFi1983b , the system implicitly appeared already in Tr1973 . In the univariate case , part a) of Theorem 1 has essentially been established by Triebel Tr1973 and Ropela Ro1976 under the restriction . Extensions to are due to Ciesielski Ci1977 and Triebel. The results of Triebel who worked in the framework of distributional Besov spaces and are summarized in (Tr2010, , Section 2). Starting from Tr1978 , Triebel considered the parameter values , , and proved the unconditionality of in for the parameter range

 max(d(1/p−1),1/p−1)

This is the essence of Theorem 2.13 (i) () and Theorem 2.26 (i) () in Tr2010 . Note that for the range we have

 Bsp,q(Id)=Bsp,q,1(Id)⟺max(0,d(1/p−1))

i.e., for the parameters in (4), the scales and coincide up to equivalent quasi-norms. Consequently, with the exception of the range for , the unconditionality of in the spaces essentially follows from Triebel’s results for for all cases stated in Theorem 1 and Theorem 3. However, we give a direct proof for using its characterization by piecewise constant best approximations on dyadic partitions.

Triebel Tr1978 also established that outside the closure of the parameter range (LABEL:PR21), the Haar wavelet system is not a Schauder basis in the spaces . The boundary cases remained unsettled until recently when Garrigós, Seeger, and Ullrich dealt in a series of papers GSU2018 ; GSU2019a ; GSU2019b ; SeUl2017a ; SeUl2017b with the open cases for both the distributional and scales. In particular, GSU2019a provides complete answers concerning the Schauder basis properties of the Haar wavelet systems in and . They also established a subtle difference between the cases and for the critical smoothness parameter ,

, and provided correct asymptotic estimates of the norms of partial sum projectors associated with

. We also mention the paper YSY2018 related to the questions considered in this paper, where the authors study necessary and sufficient conditions on the parameters for which the map extends to a bounded linear functional on Besov-Morrey-Campanato-type spaces .

Independently, for the Schauder basis property of in and spaces was also considered by the author in Os2018 . This paper expanded on Os1981 , where a partial result was stated for , namely that the univariate Haar system forms a Schauder basis in for the critical smoothness parameter if (see the remark at the end of Os1981 ). In Os1981 , it was also established that has a trivial dual if , , .

The paper is organized as follows. In Section 2, we give the necessary definitions and state auxiliary results on Besov spaces and on piecewise constant -approximation with respect to dyadic partitions. Section 3 deals with the proof of the sufficiency of the conditions on the parameters appearing in the main results formulated in Theorems 1 and 3. The necessity of these conditions and Theorem 2 are dealt with in Section 4, where we construct specific counterexamples consisting of piecewise constant functions. Similar in spirit examples have already been used in Os1981 . We conclude in Section 5 with some remarks on higher-order spline systems, analogous results for the tensor-product Haar system , and the exceptional cases and .

## 2 Definitions and auxiliary results

### 2.1 Haar systems

Recall first the definition of the univariate normalized Haar functions. By

we denote the characteristic function of a Lebesgue measurable set

, and by the univariate dyadic intervals of length , , . Then the univariate Haar system on is given by , and

 h2k−1+i=χΔk,2i−1−χΔk,2i,i=1,…,2k−1,k∈N,

Throughout the paper, we work with -normalized Haar functions. The Haar functions with can also be indexed by their supports, and identified with the shifts and dilates of a single function, the Haar wavelet . Indeed,

 hΔk−1,i:=h2k−1+i=h0(2k−1⋅−i),i=1,…,2k−1,k∈N.

The above introduced enumeration of the Haar functions is the natural ordering used in the literature, however, one can also define as the union of dyadic blocks

 H=∪∞k=0Hk,H0={h1},Hk={hΔk−1,i:i=1,…,2k−1},k∈N,

and allow for arbitrary orderings within each block . Below, we will work with the multivariate counterparts of the spaces

 Sk=span({hm}2km=1)=span({χΔk,i}2ki=1),k=0,1,…,

of piecewise constant functions with respect to the uniform dyadic partition of step-size on the unit interval .

The Haar wavelet system

 Hd=∪∞k=0Hdk (5)

on the -dimensional cube , , is defined in a blockwise fashion as follows. Let the partition be the set of all dyadic cubes of side-length in . Each cube in is the -fold product of univariate , i.e.,

 Tdk={Δk,i:=Δk,i1×…×Δk,id:i=(i1,…,id)∈{1,…,2k}d}.

The set of all piecewise constant functions on is denoted by . With each , , , we associate the set of multivariate Haar functions with support , given by all possible tensor products

 ψk,i1⊗ψk,i2⊗…⊗ψk,id,ψk,i=hΔk−1,i or χΔk−1,i

where at least one of the equals . The blocks appearing in (5) are given as follows: The block is exceptional, and consists of the single constant function . The block coincides with and consists of Haar functions, where . For general , the block

 Hdk:=∪Δk−1,i∈Tdk−1Hdk,i

consists of Haar functions of level . It is obvious that

 Sdk=span(∪kl=0Hdl),

and that is a complete orthogonal system in .

Since each Haar function in has support on a -dimensional dyadic cube, we sometimes call this system isotropic, in contrast to the anisotropic tensor-product Haar system defined in (1), where the supports of the tensor-product Haar functions are -dimensional dyadic rectangles. Note that can also be organized into blocks , where for the block consists of the tensor-product Haar functions orthogonal to , and . Obviously, we have

 span(~Hdk)=span(Hdk),k∈Z+,

such that represents a level-wise transformation of vice versa. The basis properties of in the spaces are exhaustively dealt with in Section 5.3, see Theorem 5. It turns out that for the two Haar systems and behave quite differently in this respect.

As for the univariate case, the ordering of the Haar functions within the blocks can be arbitrary. The statements of Theorems 1 and 2 hold for any enumeration of as long as the enumeration does not violate the natural ordering by level . Note that slightly more general orderings have been considered in GSU2019a for Haar wavelet systems on and .

### 2.2 Function spaces

The Besov function spaces are traditionally defined for , , as the set of all for which the quasi-norm

 ∥f∥Bsp,q,1:={(∥f∥qLp+∥t−s−1/qω(t,f)p∥qLq(I))1/q,0

is finite. Formally, the definition makes sense for all , see below for further comments in this direction. Here,

 ω(t,f)p:=sup0<|y|≤t∥Δyf∥Lp(Idy),t>0,

stands for the first-order modulus of smoothness, and

 Δyf(x):=f(x+y)−f(x),x∈Idy:={z∈Id:z+y∈Id},y∈Rd,

denotes the first-order forward difference. Here and throughout the remainder of the paper, we adopt the following notational convention: If the domain is , we omit the domain in the notation for spaces and quasi-norms, e.g., we write instead of , and instead of . An exception are the formulation of theorems. Also, by we denote generic positive constants that may change from line to line, and, unless stated otherwise, depend on only. The notation is used if holds for two such constants .

The space is a quasi-Banach space equipped with a -quasi-norm, where , meaning that is homogeneous and satisfies

 ∥f+g∥γBsp,q,1≤∥f∥γBsp,q,1+∥g∥γBsp,q,1.

Similarly, is a quasi-Banach space equipped with a -quasi-norm if we set .

If then the spaces are of interest only if . Indeed, if for some , , then using the properties of the first-order modulus of smoothness we have , , which in turn implies for all and for some constant almost everywhere on . Thus, in this case deteriorates to the set of constant functions on . On the other hand, one has which implies that

 ∥f∥Bsp,q,1≈∥f∥Lp,f∈Lp,s<0.

In other words, for all (this holds also for and ).

To conclude this short discussion of the definition and properties of -spaces, let us motivate our basic assumption (2) on the parameter range adopted in this paper. The case is in some sense exceptional and often not considered at all, we return to it in Section 5.2. Since our main concern is the Schauder basis property of the countable Haar systems and , we can also neglect all parameters for which is non-separable or does not contain piecewise constant functions on dyadic partitions. The separability requirement excludes the spaces with or . Since, with constants also depending on , we have

 ω(t,h)p≈t1/p,t→0,

for any Haar function , , we see that is not contained in whenever , . Thus, the restrictions in (2) are natural.

For completeness, we give the definition of the distributional Besov spaces using dyadic Fourier transform decompositions, see e.g.

(Tr2010, , Section 1.1). Denote by the Fourier transform operator on the set of tempered distributions. Consider a smooth partition of unity , where satisfies for and for , and for , and . Then a tempered distribution belongs to if the -quasi-norm

 ∥f∥Bsp,q(Rd):=∥(∥2ksF−1ϕkFf∥Lp(Rd))k∈Z+∥ℓq(Z+)

is finite. By exchanging the order of taking and quasi-norms in (LABEL:DiBN), one defines the Triebel-Lizorkin spaces . Spaces on domains are defined by restriction. In particular, for the domain

 Bsp,q={f:∃g∈Bsp,q(Rd) such that f=g|Id}

and

 ∥f∥Bsp,q:=infg:f=g|Id∥g∥Bsp,q(Rd).

This definition, and many equivalent ones, are surveyed in Tr2006 and (Tr2010, , Chapter 1), with references to earlier papers. In particular, the equivalence (4) is mentioned in (Tr2010, , Section 1.1).

### 2.3 Piecewise constant Lp-approximation

We start with introducing an equivalent quasi-norm in which is based on approximation techniques using piecewise constant approximation on dyadic partitions. Let

 Ek(f)p:=infs∈Sdk∥f−s∥Lp,k=0,1,…,

denote the best approximations to with respect to .

###### Lemma 0

Let , , and . Then

 ∥f∥Asp,q,1:=(∥f∥qLp+∞∑k=0(2ksEk(f)p)q)1/q (6)

provides an equivalent quasi-norm on .

This result follows from the direct and inverse inequalities relating best approximations and moduli of smoothness which have many authors. In the univariate case , see e.g. Ul’yanov Ul1964 , Golubov Go1972 for , and (SKO1975, , Section 2) for . Lemma 1 is a partial case of (DP1988, , Theorem 5.1), for and see (Os1980, , Theorem 6). The proofs for also cover the case not mentioned in these papers. Note that in DP1988 the parameter range , , is formally excluded but the result holds for the special case of piecewise constant approximation. With the appropriate modification of the quasi-norm, such an approximation-theoretic characterization also holds for and .

The norm equivalence (6) automatically implies that the set of all dyadic step functions

 Sd:=span(Hd)=span({Sdk}k∈Z+)

is dense in for the parameter values stated in Lemma 1. It can be used to prove sharp embedding theorems of into . In particular, we have continuous embeddings

 Bd(1/p−1)p,p,1⊂Bd(1/p−1)p,1,1⊂L1,(d−1)/d

We refer to Os1980 for , and to (DP1988, , Theorem 7.4) for . A local version of the associated embedding inequality will be used in Section 3.

At the heart of the counterexamples constructed in Section 4 for is a simple observation about best approximation by constants which we formulate as

###### Lemma 0

Let be a finite measure space, and let the function , , equal a constant on a measurable set of measure . Then

 ∥f−ξ0∥Lp(Ω)=infξ∈R∥f−ξ∥Lp(Ω),

i.e., best approximation by constants in is achieved by setting .

Proof. Indeed, under the above assumptions and by the inequality we have

 ∥f−ξ∥pLp(Ω) = ∫Ω′|f(x)−ξ|pdμ(x)+μ(Ω∖Ω′)|ξ−ξ0|p ≥ ∫Ω′(|f(x)−ξ|p+|ξ−ξ0|p)dμ(x) ≥ ∫Ω′|f(x)−ξ0|pdμ(x)=∥f−ξ0∥pLp(Ω)

for any , with equality for . This gives the statement.

Note that the equivalence (up to constants depending on parameters but not on ) between quasi-norms and best approximations by constants holds also for and under weaker assumptions on the relative measure of (e.g., would suffice). We will apply this lemma to the Lebesgue measure on dyadic cubes in and special examples of dyadic step functions constructed below. Extensions to higher degree polynomial and spline approximation are possible as well (see the proof of the lemma on p. 535 in Os1981 for ).

### 2.4 Schauder basis property

A sequence of elements of a quasi-Banach space is called a Schauder basis in if every possesses a unique series representation

 f=∞∑m=1cmfm

converging in . If every rearrangement of is a Schauder basis in then this system is called unconditional Schauder basis. Below we will rely on the following criterion whose proof for Banach spaces can easily be extended to the quasi-Banach space case.

###### Lemma 0

The sequence of elements of a quasi-Banach space is a Schauder basis in the quasi-Banach space if and only if its span is dense in , and there exists a sequence of continuous linear functionals on such that

 λn(fm)={1,m=n,0,m≠n,m,n∈N,

and the associated partial sum operators

 Sn(x):=n∑m=1λm(x)fm,n∈N,x∈X,

are uniformly bounded operators in .
In order for to be unconditional, all operators

 SJ(x):=∑m∈Jλm(x)fm,x∈X,

where is an arbitrary finite subset of , must be uniformly bounded operators in .

An immediate consequence of Lemma 3 is that, in order to possess a Schauder basis at all, must have a sufficiently rich dual space of continuous linear functionals.

If is a Schauder basis in a quasi-Banach space of functions defined on then must be a dense subset of by the density condition in Lemma 3. This is satisfied for all with parameters satisfying (2). Moreover, since and is an orthogonal system in , any dyadic step function has a unique Haar expansion given by

 g=∑h∈Hdλh(g)h,λh(g):=2kd∫Idghdx. (8)

Since for only finitely many coefficients do not vanish, the summation in (8) is finite, and there are no convergence issues. Thus, for the Schauder basis property of in to hold, the coefficient functionals in (8) must be extendable to elements in , and the level partial sum operators

 Pkg=k∑l=0∑h∈Hdlλh(g)h,k=0,1,…, (9)

must form a sequence of uniformly bounded linear operators in . Due to the local support properties of the Haar functions within each block and the assumed ordering of the Haar wavelet system it often suffices to deal with this subsequence of partial sum operators. The statements in Theorem 1 b) about the failure of the Schauder basis property of in will be shown by either relying on Theorem 2 or proving that the operators are not uniformly bounded.

Whenever is continuously embedded into , the level partial sum operators extend to bounded projections with range , and with constant values on the dyadic cubes in explicitly given by averaging. This comes in handy when computing for concrete functions . Indeed, the constant values taken by on dyadic cubes in are given by

 Pkf(x)=avΔ(f):=2kd∫Δfdx,x∈Δ,Δ∈Tdk,,k=0,1,…, (10)

if . Note that the coefficient functionals of the Haar expansion are finite linear combinations of functionals as defined in (10), vice versa. Finally, for the level partial sum operator realizes the orthoprojection onto .

## 3 Proofs: Sufficient conditions

### 3.1 The case s=d(1/p−1)

The positive results on the Schauder basis property of in stated in Theorem 1 b) for the parameter range

 d−1d

can be traced back to Os1981 for , we reproduce the proof for given in the preprint Os2018 . For similar results in the case of distributional Besov spaces and we refer to Tr2010 ; Os2018 ; GSU2019a .

According to (7), for the parameters in (11) we have the continuous embedding

 Bd(1/p−1)p,q,1⊂Bd(1/p−1)p,p,1⊂L1.

This ensures that the Haar coefficient functionals defined in (8) are continuous on . Moreover, is dense in . Due to Lemma 1 and Lemma 3, it is therefore sufficient to establish the inequality

for any partial sum operator of the Haar expansion (8), with a constant independent of and , if the parameters satisfy (11).

According to our ordering convention for , any partial sum operator can be written, for some and some subset , in the form

 Pg=Pkg+∑h∈¯Hdk+1λh(g)h∈Sdk+1. (13)

For , we get as partial case.

The first step for establishing (12) is the proof of the inequality

 ∥Pg∥pLp≤C2kd(p−1)∑Δ∈Tdk∥g∥pL1(Δ), (14)

with the explicit constant . By (10), we have

 ∥Pkg∥pLp=∑Δ∈Tdk2−kd(2kd∫Δgdx)p≤2kd(p−1)∑Δ∈Tdk∥g∥pL1(Δ).

The remaining can be grouped by their support cubes . Each such group may hold up to Haar functions with the same . By the definition of the Haar coefficient functionals for each term associated with such a group we obtain the estimate

 ∥λh(g)h∥pLp=|λh(g)|p∥h∥pLp(Δ)≤2kdp∥g∥pL1(Δ)⋅2−kd=2kd(p−1)∥g∥pL1(Δ).

Thus, using the -quasi-norm triangle inequality for ,

 ∥Pg∥pLp ≤ ∥Pkg∥pLp+∑h∈¯Hdk+1∥λh(g)h∥pLp ≤ 2kd(p−1)∑Δ∈Tdk∥g∥pL1(Δ)+∑Δ∈Tdk(2d−1)2kd(p−1)∥g∥pL1(Δ) = 2d2kd(p−1)∑Δ∈Tdk∥g∥pL1(Δ),

we obtain (14).

Now we apply the embedding inequality associated with (7), with the appropriate coordinate transformation, locally on to the terms . This gives

 ∥g∥pL1(Δ)≤C(2kd(1−p)∥g∥pLp(Δ)+∞∑l=k2ld(1−p)El(g)pp,Δ)

for each , where

 El(g)p,Δ:=infs∈Sdl∥g−s∥Lp(Δ),l=k,k+1,…,

denotes the local best approximation by dyadic step functions restricted to cubes from . Since

 ∥g∥pLp=∑Δ∈Tdk∥g∥pLp(Δ),E