Hyperbolic wavelet analysis of classical isotropic and anisotropic Besov-Sobolev spaces

In this paper we introduce new function spaces which we call anisotropic hyperbolic Besov and Triebel-Lizorkin spaces. Their definition is based on a hyperbolic Littlewood-Paley analysis involving an anisotropy vector only occurring in the smoothness weights. Such spaces provide a general and natural setting in order to understand what kind of anisotropic smoothness can be described using hyperbolic wavelets (in the literature also sometimes called tensor-product wavelets), a wavelet class which hitherto has been mainly used to characterize spaces of dominating mixed smoothness. A centerpiece of our present work are characterizations of these new spaces based on the hyperbolic wavelet transform. Hereby we treat both, the standard approach using wavelet systems equipped with sufficient smoothness, decay, and vanishing moments, but also the very simple and basic hyperbolic Haar system. The second major question we pursue is the relationship between the novel hyperbolic spaces and the classical anisotropic Besov-Lizorkin-Triebel scales. As our results show, in general, both approaches to resolve an anisotropy do not coincide. However, in the Sobolev range this is the case, providing a link to apply the newly obtained hyperbolic wavelet characterizations to the classical setting. In particular, this allows for detecting classical anisotropies via the coefficients of a universal hyperbolic wavelet basis, without the need of adaption of the basis or a-priori knowledge on the anisotropy.


Introduction
With the development of wavelet analysis from the beginning of the 1980s until the present time we nowadays have several powerful tools at hand to perform signal analysis with the aim to extract important information out of a signal. The information is thereby usually coded in objects easy to compute and handle -the wavelet coefficients.
Wavelet methods have been used with the known success for the purpose of compression, denoising, inpainting, classification, etc., of data, to mention just a few. Roughly speaking, the common underlying idea is the fact that a few wavelet coefficients contain a rather complete information of the signal to be analyzed. However, due to their construction principle (dyadic dilations and integer translates of a few basic "mother" functions) classical wavelets are not well-suited for the analysis of, say, anisotropic signals. In fact, a signal which is rather smooth in x-direction but rough in y-direction (such as layers in the earth, stripes on a shirt, etc.) can not be properly resolved by a classical multi-resolution analysis. The respective wavelet coefficients do not contain the anisotropic smoothness information, they rather resolve a certain minimal smoothness. That results in a bad decay of the sequence of wavelet coefficients or, in other words, a bad compression rate.
Anisotropy is not a rare phenomenon since it arises whenever physics does not act the same in different directions, e.g., geophysics, oceanography, hydrology, fluid mechanics, or medical image processing (see [4,42] among others) are some of the fields where it naturally appears. For this reason wavelets have been adapted in many different ways in order to "detect" and resolve anisotropy. There is a vast amount of literature dealing with this. For instance, there are wave atoms [13] as well as curvelets [8,9,7], shearlets [32,28,34], anisets, and anisotropic wavelets [50,51,30]. The latter concept represents a rather flexible construction since it can be build (theoretically) for any present anisotropy. The theoretical basis of anisotropic wavelet analysis is the equivalent characterization of corresponding anisotropic function spaces, like Hölder, Besov, Sobolev and Triebel-Lizorkin spaces. The major shortcoming of the existing theory is the fact that one has to know the anisotropy in advance, i.e., one has to adapt the wavelet accordingly. In other words, if physics does not provide the anisotropy parameters of the signal we are not able to resolve the signal accordingly without "trying out" several anisotropic bases. Such a method is, of course, hardly implementable in practice.
In Abry et. al. [1] it has been shown that any anisotropic Besov space -defined with respect to the cartesian axis -can "almost" be characterized with the help of the so-called hyperbolic wavelet transform. The anisotropy of the signal can then be detected using a uniform basis and is characterized by a special weight in the wavelet coefficients. This has led to an efficient algorithm for image classification and anisotropy detection applied to both synthetic and real textures (see [41,2]).
In this paper we further develop this idea of describing anisotropy with the help of the hyperbolic wavelet transform. For this reason we introduce a new family of anisotropic function spaces which are defined via a hyperbolic Littlewood-Paley analysis and for which we prove exact characterization with hyperbolic wavelets. The motivation behind this is to provide a general setting of anisotropic spaces characterized by one single basis of wavelets and thus to understand how one such fixed basis can help to describe anisotropic smoothness.
Concretely, we start with a hyperbolic Littlewood-Paley analysis defined as the usual tensor product (1) ∆j(f ) := F −1 [θ j 1 ⊗ ... ⊗ θ j d Ff ] ,j = (j 1 , . . . , j d ) ∈ N d 0 . This hyperbolic decomposition of the frequency space has been widely used for the Fourier analytic definition of the well-known spaces with dominating mixed smoothness, see [43,57] and the references therein. These spaces represent a suitable framework for multivariate appoximation, see [15,48] and the recent survey article [11]. The main ideas have been developed over more than fifty years of intense research in the former Soviet Union such that it is beyond the scope of this paper to name all the relevant references (cf. [11]).
Based on the decomposition (1), we then define spaces A s,ᾱ p,q (R d ) with A ∈ {B, F } of Besov-Lizorkin-Triebel type involving an anisotropy vectorᾱ = (α 1 , . . . , As a special case (A = F , 1 < p < ∞, q = 2), these include the Sobolev type spaces W s,ᾱ p (R d ) := F s,ᾱ p,2 (R d ), where It is important to note that the anisotropy hereby only enters in the weight 2 s j /ᾱ ∞ , where we use the short-hand notation j /ᾱ ∞ := max{j 1 /α 1 , ..., j d /α d }, but not in the choice of the Littlewood-Paley decomposition.
One of the main results of this paper is the coincidence (with respect to equivalent norms) where the space on the right-hand side represents the classical anisotropic Sobolev space defined in (3) below. This relation has already been observed for isotropic (i.e.ᾱ = (1, ..., 1)) Hilbert-Sobolev spaces (p = 2) on the d-torus, see [27,12], as well as on R 2 in [1]. Our result extends this observation to all 1 < p < ∞. Surprisingly, such a coincidence in the spirit of (2) is only possible in the Sobolev case. To be more precise, it holds A s,ᾱ p,q (R d ) = A s,ᾱ p,q (R d ) if and only if A = F , 1 < p < ∞, and q = 2.
As an important consequence of this equality (2), we can further prove that it is possible to characterize (e.g. detect and classify) classical anisotropies described by the spaces W s,ᾱ p (R d ) via the wavelet coefficients of a universal hyperbolic wavelet basis. Compared to the classical approach using anisotropic wavelets, this approach has the advantage that one does not need a-priori knowledge on the anisotropies, otherwise required for constructing the "right" basis. In particular, our results entail that any sufficiently regular orthonormal basis (ψj ,k )j ,k of tensorized wavelets ψj ,k = ψ j 1 ,k 1 ⊗ · · · ⊗ ψ j d ,k d constitutes an unconditional Schauder basis for W s,ᾱ p (R d ), whose coefficients, measured in an appropriate corresponding sequence space, give rise to an equivalent norm on W s,ᾱ p (R d ), i.e. for f ∈ W s,ᾱ p (R d ) with coefficients f, ψj ,k This is stated in Theorem 6.2. A similar result, see Theorem 6.3, holds true for the hyperbolic Haar system H d = (hj ,k )j ,k , where hj ,k = h j 1 ,k 1 ⊗ · · · ⊗ h j d ,k d , under the following restriction on the parameter s of the space W s,ᾱ p (R d ), In this direction, we would also like to mention the new and related findings of Oswald in [37] on the Schauder basis property of the hyperbolic Haar system in the classic isotropic Besov spaces defined via first-order moduli of smoothness. At the center of our respective proofs, we will rely on discrete characterizations provided by hyperbolic wavelets for the spaces A s,ᾱ p,q (R d ), A ∈ {B, F }. These characterizations are fundamental and established in separate theorems, Theorem 4.2 and Theorem 5.4, whereby we follow two paths. On the one hand, we use the usual methodology and consider orthonormal wavelet bases for which we assume sufficient smoothness, decay, and vanishing moments. As a byproduct, we thereby significantly extend the wavelet characterizations in [57,55] for Besov-Lizorkin-Triebel spaces with dominating mixed smoothness. On the other hand, we use a hyperbolic Haar system, which does not fulfill smoothness conditions as before but nevertheless allows for characterization in a certain restricted parameter range.
Let us remark that analysis with the Haar wavelet has a long tradition (see e.g. [25,38,39,40,3]), the Haar wavelet being the oldest and simplest orthonormal wavelet, conceived as early as 1909 [29]. Besides its elegance and simplicity, notably its connection to the Faber system [17] and other spline functions, such as e.g. the Chui-Wang wavelet [10], makes it interesting from a numerical perspective. In particular in imaging science it plays an important role in practical applications. Recently, it has attracted renewed attention with a series of publications [44,45,21,22,23,24,14].
The paper has the following structure. After having recalled in Section 2 some helpful Fourier analytic tools (in particular some classical maximal functions and associated inequalities) as well as the definition of the classical (anisotropic) function spaces A s,ᾱ p,q (R d ), where A ∈ {B, F }, we introduce in Section 3 the notion of hyperbolic Littlewood-Paley analysis and the related Besov-Lizorkin-Triebel spaces A s,ᾱ p,q (R d ). Wavelet characterizations of these new hyperbolic spaces are the topic of Sections 4 and 5, whereby we first resort to standard wavelets with sufficient smoothness, decay, and vanishing moments in Section 4, while in Section 5 we utilize a hyperbolic Haar basis. The relationship of the new scale to the traditional spaces is finally investigated in Sections 6 and 7. Specifically, in Section 6, we show the equality W s,ᾱ p (R d ) = W s,ᾱ p (R d ), i.e. F s,ᾱ p,2 (R d ) = F s,ᾱ p,2 (R d ), in the range 1 < p < ∞, from which we can then extract our main theorems concerning hyperbolic wavelet characterizations of the classical W s,ᾱ p (R d ). Let us agree on the following general notation. As usual N shall denote the natural numbers. We further put N 0 := N ∪ {0}, and let Z denote the integers, R the real numbers, and C the complex numbers. By T := R/2πZ we refer to the torus identified with the interval [0, 2π] ⊂ R. We write x, y or x · y for the Euclidean inner product in R d or C d . The letter d is hereby always reserved for the underlying dimension and by [d] we mean the set {1, ..., d}. For 0 < p ≤ ∞ and x ∈ R d we define x p := ( d i=1 |x i | p ) 1/p , with the usual modification in the case p = ∞. If 1 ≤ p ≤ ∞ we set p ′ such that 1/p + 1/p ′ = 1. For 0 < p, q ≤ ∞ we further denote σ p,q := max{1/p − 1, 1/q − 1, 0} and σ p := max{1/p − 1, 0}. We also put x + := ((x 1 ) + , ..., (x d ) + ), whereby a + := max{a, 0} for a ∈ R. Analogously we define x − . By (x 1 , . . . , x d ) > 0 we shall mean that each coordinate is positive. Finally, as usual, a ∈ R is decomposed into a = ⌊a⌋ + {a}, where 0 ≤ {a} < 1 and ⌊a⌋ ∈ Z. In case x ∈ R d , {x} and ⌊x⌋ are then meant component-wise. Multivariate indices are typesetted with a bar, like e.g. k,j,l, orm, to indicate the multi-index. In all the paper, the multi-indexᾱ = (α 1 , ..., α d ) > 0 thereby stands for an anisotropy and is such that α 1 + ... + α d = d. In addition, we here use the abbreviations α min := min{α 1 , .., α d } and α max := max{α 1 , .., α d }. The notation α/j shall always stand for (α 1 /j 1 , . . . , α d /j d ). Given a positive real a > 0, we further write aᾱ for the vector (a α 1 , . . . , a α d ) and let f (aᾱx) := f (a α 1 x 1 , . . . , a α d x d ) be the anisotropically scaled version of the function f : R d → C. For two (quasi-)normed spaces X and Y , the (quasi-)norm of an element x ∈ X will be denoted by x X . The symbol X ֒→ Y indicates that the identity operator is continuous. For two sequences a n and b n we will write a n b n if there exists a constant c > 0 such that a n ≤ c b n for all n. We will write a n ≍ b n if a n b n and b n a n .

Classical spaces and tools from Fourier analysis
with the usual modification if p = ∞. We will also need L p -spaces on compact domains Ω ⊂ R d instead of R d and shall write f Lp(Ω) for the corresponding restricted L p -(quasi-)norms.
For k ∈ N 0 , we denote by C k 0 (R d ) the collection of all compactly supported functions ϕ on R d which have uniformly continuous derivatives Dγϕ on R d whenever γ 1 ≤ k. Additionally, we define the spaces of infinitely differentiable functions C ∞ (R d ) and infinitely differentiable functions with compact support C ∞ 0 (R d ) as well as the Schwartz space S = S(R d ) of all rapidly decaying infinitely differentiable functions on R d , i.e., The space S ′ (R d ), the topological dual of S(R d ), is also referred to as the space of tempered distributions on R d . Indeed, a linear mapping f : S(R d ) → C belongs to S ′ (R d ) if and only if there exist numbers k, ℓ ∈ N and a constant c = c f such that The space S ′ (R d ) is equipped with the weak * -topology.
For f ∈ L 1 (R d ) we define the Fourier transform and the corresponding inverse Fourier transform F −1 f (ξ) = Ff (−ξ). As usual, the Fourier transform can be extended to The convolution ϕ * ψ of two square-integrable functions ϕ, ψ is defined via the integral If ϕ, ψ ∈ S(R d ) then ϕ * ψ still belongs to S(R d ). In fact, we have ϕ * ψ ∈ S(R d ) even if ϕ ∈ S(R d ) and ψ ∈ L 1 (R d ). The convolution can be extended to , which makes sense pointwise and is a C ∞ -function on R d .
with the usual modification for q = ∞. This definition does not depend on chosen resolution of unity ϕᾱ 0 and the quantity is a norm (resp. quasi-norm) on B s,ᾱ p,q (R d ) for 1 ≤ p, q ≤ ∞ (resp. 0 < min{p, q} < 1) and with the usual modification if q = ∞.
As in the isotropic case, anisotropic Besov spaces encompass a large class of classical anisotropic function spaces (see [51] for details). For example, when p = q = 2, the Besov spaces coincide with the anisotropic Sobolev spaces and, when p = q = ∞, the spaces B s,ᾱ ∞,∞ (R d ) are called anisotropic Hölder spaces and are denoted by C s,ᾱ (R d ).
with the usual modification for q = ∞. This definition does not depend on the chosen resolution of unity ϕᾱ 0 and the quantity 2 jsq |∆ᾱ j f (·)| q 1/q p is a norm (resp. quasi-norm) on F s,ᾱ p,q (R d ) for 1 ≤ p < ∞ and 1 ≤ q ≤ ∞ (resp. 0 < min{p, q} < 1) and with the usual modification if q = ∞.
If q = 2 and 1 < p < ∞, the anisotropic Triebel-Lizorkin space coincides with the anisotropic Sobolev space denoted by W s,ᾱ p (R d ) : Remark 2.1 (i) As mentioned before, ifᾱ = (1, ..., 1), it is easy to check that the spaces B s,ᾱ p,q (R d ) (resp. F s,ᾱ p,q (R d )) coincide with the classical spaces B s p,q (R d ) (resp. F s p,q (R d )). In addition, we have F 0,ᾱ p,2 (R d ) = L p (R d ) in the range 1 < p < ∞. (ii) Our understanding of anisotropic spaces coincides with the one in Triebel [51] (see also the references therein). There are different (but related) notions of anisotropy in the Russian literature, see Nikolskij [36,Chapt. 4] or Temlyakov [48,II.3]. A consequence of our Theorem 6.1 below is the fact that in case of W -spaces the mentioned approaches coincide and lead to the same notion of anisotropy. However, in case of Hölder-Nikolskij spaces this is in general not the case as for instance Theorem 7.1 shows.

Maximal inequalities
Let us provide here the maximal inequalities for the Hardy-Littlewood and Peetre maximal functions, respectively. For further details we refer to [57, 1.2, 1.3] or [43,Chapt. 2] .
For a locally integrable function f : R d → C we denote by M f (x) the Hardy-Littlewood maximal function defined by where the supremum is taken over all cubes with sides parallel to the coordinate axes containing x. A vector valued generalization of the classical Hardy-Littlewood maximal inequality is due to Fefferman and Stein [18]. We require a direction-wise version of (4) 1 2s x i +s We The following construction of a maximal function is due to Peetre, Fefferman, and Stein. Let b = (b 1 , ..., b d ) > 0, a > 0, and f ∈ L 1 (R d ) with Ff compactly supported. We define the Peetre maximal function Pb ,a f by

Lemma 2.1
Let Ω ⊂ R d be a compact set. Let further a > 0 andγ = (γ 1 , ..., γ d ) ∈ N d 0 . Then there exist two constants c 1 , c 2 > 0 (independent of f ) such that holds for all f ∈ L 1 (R d ) with supp (Ff ) ⊂ Ω and all x ∈ R d . The constants c 1 , c 2 depend on Ω.
We finally give a vector-valued version of the Peetre maximal inequality which is a direct consequence of Lemma 2.1 together with Theorem 2.2.
is compact for ℓ ∈ I. Then there is a constant C > 0 (independent of f and Ω) such that

Hyperbolic Littlewood-Paley analysis
Let θ 0 ∈ S(R) be supported on [−2, 2] with θ 0 = 1 on [−1, 1]. For any j ∈ N, let us further define such that (θ j ) j is a univariate resolution of unity, i.e., j≥0 θ j (·) = 1. Observe that, for any Let us now come to the main concept of this paper, the hyperbolic Littlewood-Paley analysis.
The function θj belongs to S(R d ) for allj ∈ N d 0 and is compactly supported on a dyadic rectangle. Further j ∈N d 0 θj ≡ 1 and (θj)j is called a hyperbolic resolution of unity.
is called a hyperbolic Littlewood-Paley analysis of f .
We are now in the position to introduce new functional spaces called anisotropic hyperbolic Besov spaces and anisotropic hyperbolic Triebel-Lizorkin spaces defined via the hyperbolic Littlewood-Paley analysis.
with the usual modification in case q = ∞. This definition does not depend on the chosen resolution of unity (θj)j and the quantity is a norm (resp. quasi-norm) on B s,ᾱ p,q (R d ) for 1 ≤ p, q ≤ ∞ (resp. 0 < min{p, q} < 1) and with usual modification if q = ∞.
with the usual modification in case q = ∞. This definition does not depend on the chosen resolution of unity (θj)j and the quantity

Remark 3.2
The above definitions of anisotropic hyperbolic Besov and Sobolev spaces include four indices: s stands for the regularity, p is the integration parameter and q the so-called fine-index. The parameterᾱ = (α 1 , . . . , α d ) encodes the present anisotropy: the more α min = min{α 1 , .., α d } is close to 0 and α max = max{α 1 , ..., α d } is close to d, the more we need directional smoothness in one axis compared to others. On the other hand, ifᾱ = (1, ..., 1) the anisotropy becomes an "isotropy".

Remark 3.3
By analogy with the classical spaces, if q = 2 and 1 < p < ∞, F s,ᾱ p,q (R d ) is called anisotropic hyperbolic Sobolev space and is denoted by W s,ᾱ p (R d ). In caseᾱ = (1, ..., 1) we write W s p (R d ).
Let us finally introduce classical spaces with dominating mixed smoothness in the spirit of [43,57].
Then, there is a positive constant C > 0 such that

Hyperbolic wavelet analysis
In this section we prove hyperbolic wavelet characterizations of the spaces B s,ᾱ p,q (R d ) and F s,ᾱ p,q (R d ) defined in Definitions 3.2 and 3.3, respectively. It should be noted that the proof technique used for Theorem 4.2 below also represents a progress towards new optimal wavelet characterizations of Besov-Lizorkin-Triebel spaces with dominating mixed smoothness, which extends the results in [57,Sect. 2.4] significantly, see Remark 4.3 below.
Let us start with univariate wavelets given by a scaling function ψ 0 and a corresponding wavelet ψ. These functions are supposed to satisfy the following (minimal) conditions: (L) The wavelet ψ has vanishing moments up to order L − 1: In case L = 0 the condition is void.
and ψ 0,k := ψ 0 (· − k). We set ψ j,k ≡ 0 if j < 0. To obtain the hyperbolic wavelet basis in L 2 (R d ) we tensorize over all scales and obtain The following lemma recalls a useful convolution relation. Let us clarify the notation first. For a given univariate function Λ we will use the notation Λ j (·) := 2 j−1 Λ(2 j−1 ·), j ∈ N . We will further put x j,m := 2 −j m and I j,m := [2 −j m, 2 −j (m + 1)) with associated characteristic function χ j,m := ½ I j,m .
Lemma 4.1 Let Λ 0 , Λ ∈ S(R) with Λ having infinitely many vanishing moments, i.e., for all β ∈ N . Let further ψ 0 and ψ satisfy (K) and (L) as above and R > 0 be a given real number. Then it exists a constant C R > 0 such that for any j ∈ N 0 and ℓ, m ∈ Z the convolution relation Proof. The above lemma is a special case of a more general convolution relation, see for instance [26, p. 466 Lemma 4.1 immediately implies the following multivariate version by exploiting the tensor product structure. Similar as for the hyperbolic wavelet system, we use the notation In the sequel we will further need the notation with the notation I j i ,m i and χ j i ,m i , i ∈ [d] = {1, . . . , d}, introduced right before Lemma 4.1.
Lemma 4.2 Let Λ, Λ 0 , ψ 0 , ψ as in Lemma 4.1. For any R > 0 there exists a contant C R > 0 such that for anyj ∈ N d 0 andl,m ∈ Z d the convolution relation The next proposition is also crucial and represents the "hyperbolic version" of [31,Lem. 3,7]. An isotropic version is originally due to Kyriazis [33,Lem. 7.1]. For the convenience of the reader we give a proof.
where M stands for the Hardy-Littlewood maximal operator.
Proof. We follow the proof from [31,Lem. 7]. Put δ = R − 1/r > 0 and define a decompo- with We then estimate for fixed We further note that |λj +l,w | r χj +l,w (y) dy and observe that for Q(x) :=

m∈Ωk(x)
Qj +l,m we have x ∈ Q(x) and Recalling the definition of the Hardy-Littlewood maximal function in (4), we obtain Putting this into (10), we arrive at Finally, we plug this estimate into (9) and obtain the desired assertion.
Before stating our main result we need a further definition.
(i) If 0 < p < ∞ we define the sequence spacef s,ᾱ p,q as the collection of all sequences (ii) If 0 < p ≤ ∞ we define the sequence spaceb s,ᾱ p,q as the collection of all sequences Now we are ready to state the wavelet characterization of the space F s,ᾱ p,q (R d ) . Recall that for 0 < p, q ≤ ∞ we put σ p,q := max{1/p − 1, 1/q − 1, 0} and σ p := max{1/p − 1, 0} .

Remark 4.1
The theorem below states the result for the F -scale of spaces F a,ᾱ p,q (R d ). As for the corresponding result for the Besov type spaces B s,ᾱ p,q (R d ), we simply replace condition (11) on K, L by K, L > σ p + |s|/α min and use the corresponding sequence spacesb s,ᾱ p,q .
if and only if it can be represented as with (λj ,k )j ,k ∈f s,ᾱ p,q and the sum converging in S ′ (R d ) with respect to some ordering. For each f ∈ F s,ᾱ p,q (R d ) the convergence of the representation (12) is then even unconditional. Moreover, if q < ∞, the sum also converges in F s,ᾱ p,q (R d ) and (ψj ,k )j ,k constitutes an unconditional basis in F s,ᾱ p,q (R d ). The sequence of coefficients λ(f ) := (λj ,k )j ,k is uniquely determined via and we have the wavelet isomorphism (equivalent (quasi-)norm) Remark 4.2 Following [35,Prop. 3.20], the dual pairing of f ∈ S ′ (R d ) and ψj ,k ∈ C K (R d ) in (13) has to be understood in the way Here we choose Θj := F −1 θj and Λj := F −1 λj such that ∞ j=0 θ j λ j ≡ 1 . Using elementary estimates and the Nikolskij inequality in case p < 1, one can show Setting sᾱ ,p := |s|/α min + σ p andp := max{p, 1} we obtain where the right-hand side is finite due to (11) and (8).
In other words, f ∈ F s,ᾱ p,q (R d ) generates a (conjugate) linear functional on the Banach space S sᾱ,p p ′ ,1 B(R d ).
Remark 4.3 As we will see below, our arguments apply as well to classical spaces of dominating mixed smoothness S r p,q B(R d ) and S r p,q F (R d ), defined in Definition 3.4 above. Examining the proof, we obtain for the relation where s r p,q f is the sequence space associated to S r p,q F (for a definition see [57, Def. 2.1]), the condition (15) L > σ p,q − r and K > r .
The converse relation holds under the condition K > σ p,q − r and L > r .
For the spaces S r p,q B(R d ) we replace σ p,q by σ p and s r p,q f by s r p,q b, which is the sequence space associated to S r p,q B (for a definition see [57, Def. 2.1]).

Proof. [of Theorem 4.2]
Step 1. We consider the sum with λ := (λj ,k )j ,k ∈f s,ᾱ p,q and show the relation For the issues on the convergence and uniqueness of (16) and (13) we refer to Step 3 and 4 below, where we show that under the assumption (λj ,k )j ,k ∈f s,ᾱ p,q the element f is well defined, with unconditional convergence of (16) at least in S ′ (R d ), which is sufficient for the subsequent considerations.
Let us consider ∆j(f ) for some chosen hyperbolic Littlewood-Paley analysis. This gives for fixedj ∈ N d where Θj := F −1 θj and (θj)j is the system from Definition 3.1. With u := min{p, q, 1} With the help of Lemma 4.2 we are aiming for pointwise estimates first.
Let us prove the converse relation λ(f ) f s,ᾱ p,q f F s,ᾱ p,q (R d ) with λ(f ) = (2 j 1 f, ψj ,k )j ,k and start with f ∈ F s,ᾱ p,q (R d ). As already pointed out in Remark 4.2, the dual pairing f, ψj ,k makes sense due to condition (11). Our estimation begins as follows where we put . We then obtain for z = (z 1 , . . . , z d ) ∈ Qj ,k the estimate This together with (19) and Proposition 4.1 yields for r < min{p, q, 1} = u and z ∈ R d This leads to Taking the L p (ℓ q [j])-(quasi-)norm on both sides and using (18) once more, we obtain Due to r < min{1, p, q} = u we can apply the Hardy-Littlewood maximal inequality (Theorem 2.1) and obtain From (20) we obtain for any z ∈ Qj ,m and any a > 0 where we used the definition of the Peetre maximal function in (5). Choosing a > max{ 1 p , 1 q } with the corresponding maximal inequality in Theorem 2.3 then yields the relation Returning to (21), we have seen It remains to discuss the sum overl. It is easy to see that it converges if K + 1 = M + > 1/r + |s|/α min and L = M − > |s|/α min . Recall that r is chosen such that r < min{1, p, q}.
Step 3. Let us now clarify the convergence issues in (12) in case q < ∞. The arguments in Step 1 above show in particular for a finite partial summation of (12) that If q < ∞ (note that p < ∞ anyway) we use Lebesgue's dominated convergence theorem to conclude the unconditional convergence of (12) in F s,ᾱ p,q (R d ). The required majorant is thereby given by In case q = ∞ we use the observation in Remark 4.3. From a simple application of Hölder's inequality (with respect to the sum overj) we first obtain for any ε > 0 the relation with r(s,ᾱ) := −|s|/α min .
Choosing ε > 0 small enough, we then obtain from (11) that condition (15) in Remark 4.3 is satisfied. Hence, for a finite partial summation of (12) we have Again, by Lebesgue's dominated convergence theorem (the majorant given by (22)) we see the unconditional convergence of (12) in the space S r(s,α)−ε p,1 into account, we actually proved more than stated in the theorem.
Step 4. It remains to prove (12) for f ∈ F s,ᾱ p,q (R d ) and coefficients λj ,k (f ) chosen as in (13). From Steps 1, 2, 3 above we have learned that {λj ,k (f )}j ,k ∈f s,ᾱ p,q , which implies that the sum converges (at least in) S ′ (R d ) to an element g ∈ S ′ (R d ). We now prove that f (ϕ) = g(ϕ) for all ϕ ∈ S(R d ). Fix ϕ ∈ S(R d ), then clearlyφ ∈ L 2 (R d ) and we have with convergence in L 2 (R d ). Sinceφ ∈ S sᾱ,p p ′ ,1 B(R d ) we have by Step 1, 2, 3 above that the right-hand side of (23) converges in S sᾱ,p . Hence, we have φ = η in S ′ (R d ) which finally givesφ = η almost everywhere and, in other words, (23) holds true in S sᾱ,p p ′ ,1 B(R d ). Then f (ϕ) can be rewritten as follows, using the continuity of f, · (see Remark 4.2), On the other hand, which finishes the proof.

Hyperbolic Haar characterization
We next utilize a hyperbolic Haar basis for the characterization of the spaces B s,ᾱ p,q (R d ) and F s,ᾱ p,q (R d ) from Definitions 3.2 and 3.3, the main result being Theorem 5.4. It will show that Haar characterizations are possible in a certain restricted range of parameters, although the Haar wavelet does not fulfill smoothness requirements (K) as assumed for the derivation of Theorem 4.2 in the previous section. Hence, for the proof of Theorem 5.4 a different methodology is needed than in Section 4. We follow the technique used in [23], exploiting the special structure of the Haar wavelet.
We begin by fixing a convenient inhomogeneous Haar system on the real line, namely where for j ∈ N, k ∈ Z, the functions h j,k are scaled Haar functions of the form The intervals I + j,k = [2 −j k, 2 −j (k +1/2)) and I − j,k = [2 −j (k +1/2), 2 −j (k +1)) thereby represent the dyadic children of the standard dyadic intervals I j,k = [2 −j k, 2 −j (k + 1)). At the lowest scale j = 0 the ordinary Haar functions ½ I + 0,k −½ I − 0,k are replaced by the characteristic functions h 0,k := ½ I 0,k . Further, we set h j,k ≡ 0 if j < 0. Defined like this, the structure of the system H 1 fits closely to the wavelet systems considered in Section 4. The inhomogeneous scale is at j = 0 (and not the usual standard j = −1 for Haar systems).
For dimension d ∈ N we derive a corresponding hyperbolic d-variate Haar system by the following tensorization procedure, Note that forj ∈ N d andk ∈ Z d the cube whose characteristic function will subsequently be denoted by χj ,k as already earlier in (7), corresponds to the strict support of the Haar function hj ,k . At each fixed "scale"j these cubes represent a partition of the d-dimensional domain R d .
we have for f ∈ S ′ (R d ) (with the dual pairing f, hj ,k defined as in (14) in Remark 4.2) whenever the left-hand side is defined. In case |s|/α min < 1 − 1 p we have Proof. For the proof, we first build a suitable decomposition of unity adapted to the hyperbolic tiling of the frequency domain. For a respective construction, we start with univariate functions φ 0 , φ ∈ S(R) and λ 0 , λ ∈ S(R) such that where φ j := φ(2 −j ·) and λ j := λ(2 −j ·) for j ∈ N. The functions φ 0 and φ 1 shall thereby, as usual, be compactly supported with for some ε > 0. As a consequence, their inverse Fourier transforms however, namely Φ 0 := F −1 φ 0 and Φ := F −1 φ, cannot have compact supports.
The functions λ 0 , λ, on the other hand are chosen such that the supports of Λ 0 := F −1 λ 0 and Λ := F −1 λ are compact. Further, they are assumed to fulfill the Tauberian conditions with the same ε > 0 as above and furthermore Dγλ(0) = 0 for multi-indicesγ ∈ N d 0 with γ 1 ≤ 1. Such a construction is indeed possible, see [53, Lem. 3.6] for example.
For the subsequent proof, it is convenient to also define the functions Φ j := F −1 φ j and Λ j := F −1 λ j for j ∈ N. They fulfill the scaling relations Φ j = 2 j Φ(2 j ·) and Λ j = 2 j Λ(2 j ·).
Next, we put Φ j := 0 and Λ j := 0 for j ∈ Z with j < 0 and build the tensor products Φl := i∈{1,...,d} Φ ℓ i and Λl := i∈{1,...,d} Then we have the decomposition, which in fact is a discrete version of Calderón's reproducing formula, enabling a component-wise evaluation of the scalar product f, hj ,k . Each Haar coefficient can in this way be understood in the following sense (see also Remark 4.2), whenever the right-hand sum converges.
If we further assume that Λl is even, we arrive at the estimate Let us investigate the integral on the right-hand side, and for this let us define Sj ,k,l := supp Λj +l * hj ,k .
If min i∈[d] {j i + ℓ i } < 0 we have Sj ,k,l = ∅ and the integral vanishes. Otherwise, when min i∈[d] {j i + ℓ i } ≥ 0, we fix a > 0 and x ∈ R d and obtain the estimate where P 2j +l ,a (Φj +l * f ) denotes the Peetre maximal function (see (5))

The integral term splits into
according to the relation For fixed i ∈ {1, . . . , d}, assuming ℓ i < 0, we can then further estimate For the latter of these two inequalities the first order vanishing moment of the Haar wavelet comes into play. Note here that indeed j i > 0 due to min i∈ [d] {j i + ℓ i } ≥ 0, allowing for the estimate In case ℓ i ≥ 0 we obtain a different estimate than (27), namely Here we use the fact that the integrand is bounded by a constant together with the observation that its support is contained in at most three intervals of length ≍ 2 −(j i +ℓ i ) . Indeed, as a consequence of the L 1 -resp. L ∞ -normalization of Λ j i +ℓ i and h Furthermore, due to the vanishing moment properties of Λ j i +ℓ i , the support of the convolution merely stems from the either two or three discontinuities of the function h j i ,k i . Now, let us turn our attention to the factor Here, we have with Putting all together, this yields for x ∈ Qj ,k Hence, we obtain uniformly in x ∈ R d and for fixedj ∈ N d Finally, we can turn to the proof of (25). We estimate According to (18) it holds 2 ( j /ᾱ ∞− (j+l)/ᾱ ∞)s ≤ 2 l /ᾱ ∞|s| for s ∈ R, and hence where Young's convolution inequality was used in the last step. For (26) we argue analogously, namely Choosing a > max{1/p, 1/q} in the F-case and a > 1/p in the B-case, to ensure the boundedness of the Peetre maximal operator (see Theorem 2.3, also compare e.g. [54,Thm. 2.6]), as well as |s| < min i∈{1,...,d} and thus (25) and (26), respectively.
Using a duality argument, we can deduce an immediate companion result.
we have for all f ∈ S ′ (R d ) (with the dual pairing f, hj ,k defined as in (14) in Remark 4.2) whenever the right-hand side is defined. In case |s|/α min < 1/p we have Proof. We showed in Proposition 5.1 (i) that the linear operator is well-defined and bounded in the parameter range |s| < min i∈{1,...,d} Consequently, in this range, the dual operator is also well-defined and bounded. Identifying f s,ᾱ p,q ′ withf −s,ᾱ p ′ ,q ′ with respect to the nonstandard duality product which is possible according to Theorem 5.5 (i) below, it can be represented in the form where the convergence is weak*ly in F s,ᾱ p,q ′ . This is a consequence of the relation where we used the short-hand notation Y =f s,ᾱ p,q and X = F s,ᾱ p,q . Invoking Theorem 5.5(i) another time, X ′ can be identified with F −s,ᾱ p ′ ,q ′ (and X ′′ with F s,ᾱ p,q ). Then for (λj ,k )j ,k ∈f −s,ᾱ p ′ ,q ′ and some enumeration Hence, A ′ (λ n ) n≤N → A ′ (λ n ) n∈N strongly and unconditionally in F −s,ᾱ p ′ ,q ′ . In other words, we have shown that if |s| < min i∈{1,...,d} {α i } min{1/p, 1/q} and (λj ,k )j ,k ∈f s,ᾱ p,q then j ,k λj ,k hj ,k converges strongly and unconditionally in F s,ᾱ p,q and j ,k Hence, choosing λj ,k := 2 j 1 f, hj ,k and assuming a finite sequence norm (λj ,k )j ,k f s,ᾱ p,q , then j ,k λj ,k hj ,k = j ,k 2 j 1 f, hj ,k hj ,k converges strongly to some limitf ∈ F s,ᾱ p,q (R d ). Since this sum also converges (weak*ly) in S ′ (R) to f we have f =f = j ,k 2 j 1 f, hj ,k hj ,k in F s,ᾱ p,q . Now (28) follows from (31). The proof of (29) works the same, using Proposition 5.1 (ii) and Theorem 5.5 (ii).
Combining both, Proposition 5.1 and Proposition 5.2, we arrive at the following proposition, whereby we now concentrate on the F-case.
we have for all f ∈ S ′ (R d ) (with the dual pairing f, hj ,k defined as in (14) in Remark 4.2) whenever the left-hand side is defined.
As a direct consequence of this result, we can finally formulate the main theorem of this section which corresponds to Theorem 4.2.
Then the Haar system H d = (hj ,k )j ,k defined in (24) constitutes an unconditional basis of F s,ᾱ p,q (R d ) with associated sequence spacef s,ᾱ p,q . The unique sequence of basis coefficients for Further, we have the wavelet isomorphism (equivalent norm) In addition, we can use H d to distinguish those elements of S ′ (R d ) that belong to F s,ᾱ p,q (R d ). Those are characterized by either of the following two criteria: (i) f can be represented as a sum with coefficients (λj ,k )j ,k ∈f s,ᾱ p,q (with respect to some chosen ordering).
In both cases, the sequence (λj ,k )j ,k is necessarily the sequence of basis coefficients and the representation (35) converges unconditionally to f in F s,ᾱ p,q (R d ).
Proof. As a direct consequence of the equivalence (32) proved in Proposition 5.3 the analysis operator λ : f → 2 j 1 f, hj ,k j ,k is well-defined and bounded from F s,ᾱ p,q tof s,ᾱ p,q . Moreover, it is injective and we have the equivalence of norms f F s,ᾱ p,q ≍ λ(f ) f s,ᾱ p,q . Further, for f ∈ S ′ (R d ) we have f ∈ F s,ᾱ p,q if and only if λ(f ) ∈f s,ᾱ p,q (whenever λ(f ) is defined). Now, let us have a look at the synthesis operator Clearly, for every finite sequence the assignment S defines an element in F s,ᾱ p,q . By completion, using (32) and the fact that the finite sequences lie dense inf s,ᾱ p,q , this synthesis further extends to all sequences off s,ᾱ p,q , with unconditional and strong convergence of (36) in F s,ᾱ p,q . Hence, S is a well-defined bounded linear operator fromf s,ᾱ p,q to F s,ᾱ p,q , which, as another consequence of (32), is also injective.
Next, turning to the composition λ • S operating fromf s,ᾱ p,q tof s,ᾱ p,q , we deduce for each using the orthogonality of the system (hj ,k )j ,k in L 2 (R d ). We obtain Idfs,ᾱ p,q = λ•S and in turn λ • S • λ = λ. Due to the injectivity of λ, the latter equality further implies Id F s,ᾱ p,q = S • λ. In particular, λ and S are thus bijections and every f ∈ F s,ᾱ p,q allows for a representation (35). To see that the representing coefficients (λj ,k )j ,k are unique, under the assumption of strong convergence of the sum, let λ * = (λ * j,k )j ,k be some sequence which satisfies (35) in a strong sense for some special ordering of the sum. Then again (32) together with a completion argument yields λ * ∈f s,ᾱ p,q , and thus λ * = λ(f ) by the injectivity of S. Hence, the expansion coefficients in (35) are unique and it follows that H d is a basis. Its unconditionality is due to the fact that the convergence of (35) for sequences λ * ∈f s,ᾱ p,q is always unconditional. For the proof of criterion (i) we just remark that for sequences (λj ,k )j ,k inf s,ᾱ p,q with weak*convergence of (35) in S ′ (R d ) the convergence is automatically in the stronger sense of F s,ᾱ p,q .
Remark 5.1 For brevity, the above theorem was only stated for the F-case. There also exists a B-version, which reads precisely the same apart from condition (33) which is replaced by In the proof of Proposition 5.2 we utilized isomorphisms F s,ᾱ p,q Hence, for the completeness of our exposition, it remains to establish those.
whereby the second of these isomorphies has to be understood with respect to the non-standard pairing (30). In the Besov case, we have the analogous relations The proof of this theorem is based on two auxiliary propositions. The first one of these is stated without proof, since it is a straightforward generalization of the classic identifications L p (R d ) ′ ∼ = L p ′ (R d ) and ℓ p (I) ′ ∼ = ℓ p ′ (I) when 1 < p < ∞ (see e.g. [16]).
Proposition 5.6 Let I be an arbitrary countable index set. Then for the respective cases Y = ℓ q I, L p (R d ) and Y = L p R d , ℓ q (I) .
The second proposition provides an alternative way to characterize functions in F s,ᾱ p,q (R d ) and B s,ᾱ p,q (R d ). Its counterpart in the classical setting of Triebel-Lizorkin spaces is Proposition 1 in [52,Sec. 2.3.4]. Proof.
as well as This settles one direction of the assertion. For the other direction, let f ∈ S ′ (R d ) satisfy (37) where the last two lines are due to the Hörmander-Mikhlin multiplier theorem, which is applied twice. This estimate shows f ∈ B s,ᾱ p,q , finishing the proof. Now we are ready to give a thorough proof of the duality relations stated in Theorem 5.5.
Proof. [of Theorem 5.5] We restrict to the F-case and begin with the more involved relation F s,ᾱ p,q The subsequent proof is thereby an adaption of the proof of the classical theorem in [52,Sec. 2.11.2] to the setting of hyperbolic spaces.
It is essential to note that, since S(R d ) lies dense in F s,ᾱ p,q , there is a natural embedding Hence, both F s,ᾱ p,q ′ and F −s,ᾱ p ′ ,q ′ can be interpreted as subspaces of S ′ (R d ), simplifying the following considerations.
Let us first assume that f ∈ S ′ (R d ) is an element of F −s,ᾱ p ′ ,q ′ and take Φl and Λl as in the proof of Proposition 5.1, for instance. Then f defines an element of F s,ᾱ as can be seen by the following estimate, Hereby, we applied Hölder's inequality and used Φl = Λl = 0 ifl / ∈ N d 0 . The duality product ·, · * thus yields an embedding ι : F −s,ᾱ p ′ ,q ′ → F s,ᾱ p,q ′ . Further, we have natural embeddings ν : F −s,ᾱ p ′ ,q ′ ֒→ S ′ (R d ) and κ : F s,ᾱ p,q ′ ֒→ S ′ (R d ) (see (38)). To establish a bridge between κ, ι, and ν, we now consider the special case of a Schwartz function g = φ ∈ S(R d ) in (39). We obtain Hence, the above (somewhat artificially) defined operation of f on F s,ᾱ p,q via ·, · * is compatible with the operation of f as an element of S ′ (R d ) on S(R d ). This proves ν = κ • ι and thus It remains to prove the converse inclusion F s,ᾱ p,q ′ ⊂ F −s,ᾱ p ′ ,q ′ . For this, let f ∈ S ′ (R d ) be an element of F s,ᾱ p,q ′ . We will show that this implies f ∈ F −s,ᾱ p ′ ,q ′ and to this end start with a construction of an isometric embedding Thereby, we build upon the observation that the assignment g → (2 |j/ᾱ|∞s ∆jg)j maps F s,ᾱ p,q isometrically to a closed subspace of L p R d , ℓ q . Via this assignment and the Hahn-Banach extension theorem, it is therefore possible to identify each functional f ∈ F s,ᾱ p,q ′ with a functional on L p R d , ℓ q having the same norm. Invoking Proposition 5.6 (i), this then yields an associated family (fj)j ∈ L p ′ R d , ℓ q ′ with (fj)j L p ′ (R d ,ℓ q ′ ) = f ( F s,ᾱ p,q ) ′ and f, g = j fj, 2 |j/ᾱ|∞s ∆jg , establishing (40).
In particular, for every φ ∈ S(R d ) In view of Proposition 5.7 (i), this shows f ∈ F −s,ᾱ p ′ ,q ′ and finishes the proof of F s,ᾱ p,q ′ ∼ = F −s,ᾱ p ′ ,q ′ . We next establish f s,ᾱ p,q ′ ∼ =f −s,ᾱ p ′ ,q ′ , which can be elegantly done using the previous result together with the wavelet isomorphism λ : F s,ᾱ p,q →f s,ᾱ p,q established in Theorem 4.2. For this, we first verify that λ preserves the duality structure of F −s,ᾱ p ′ ,q ′ × F s,ᾱ p,q . Indeed, for f ∈ F −s,ᾱ p ′ ,q ′ and g ∈ F s,ᾱ p,q we have Note that hereby we relied on the strong convergence of the wavelet expansion in the space F −s,ᾱ p ′ ,q ′ . Next we recall the isomorphism ι :

Hyperbolic and classical (anisotropic) Sobolev spaces
In the remaining two sections we will analyze the relationship between the newly introduced hyperbolic scale of spaces A s,ᾱ p,q (R d ) from Section 3, where A ∈ {B, F }, and the classical scale of anisotropic spaces A s,ᾱ p,q (R d ), which was recalled in Section 2. Our first result shows that, surprisingly, for Sobolev spaces (i.e. the case A = F , 1 < p < ∞, q = 2) both scales coincide. Theorem 6.1 Let 1 < p < ∞, s ∈ R, andᾱ > 0 be an anisotropy vector as above. Then (in the sense of equivalent norms).
Proof. The proof is divided into two steps. For convenience, we will thereby abbreviate by mᾱ ,s : the function which appears in the definition (3) of W s,ᾱ p .
Step 1. In the first step we prove f W s,ᾱ p f W s,ᾱ p . Forj = (j 1 , ..., j d ) ∈ N d 0 , let (ϕj)j denote a fixed hyperbolic resolution of unity as introduced in Definition 3.1, with corresponding univariate family (ϕ j ) j where supp (ϕ 0 ) ⊂ [−2, 2]. In addition, let us also construct a second hyperbolic resolution of unity (ψj)j such that ψjϕj = ϕj for everyj ∈ N d 0 . Hereby, it is not possible for (ψj )j to obey the same strict building law as formulated in Definition 3.1. We define functions and then put ψ j := ψ * j /3 for j ∈ N 0 . Then clearly j ψ j = 1 and ψ j ϕ j = ϕ j . Finally, we set ψj := ψ j 1 ⊗ · · · ⊗ ψ j d to obtain (ψj)j .
Step 2. For the proof of the converse inequality f W s,ᾱ p f W s,ᾱ p we argue analogously to Step 1 and use this time the multiplier It is well-defined since mᾱ ,s > 0, and we have, using the same notation as in Step 1, .
Remark 6.1 We mention that, in contrast to this result, in case A = B we only have coincidence when p = q = 2. A proof can be found in [1].
As a direct consequence of Theorem 6.1 and Theorem 4.2, we obtain new characterizations of classical Sobolev spaces via hyperbolic wavelets.
Then any f ∈ S ′ (R d ) belongs to W s,ᾱ p (R d ) if and only if it can be represented as . This, in turn, implies ϕᾱ j = 1 on the subset I α 1 j × · · · × I α d−1 j × J α d j . Observe now that for every ℓ ∈ N and every i ∈ {1, . . . , d} either is a nonempty interval of nonzero length. This is due to the fact that always where L(I) and R(I) denote the left resp. right endpoint of a given interval I = [a, b] and γ = α 2 min /8. The verification of this fact is postponed to Step 3 at the end of this proof. As a consequence, for each i ∈ {1, . . . , d} and each ℓ ∈ N, we may pick one of those intersections with nonvanishing interior and denote it byĨ (i) ℓ . Depending on our choice, we then either haveĨ Due to the nonvanishing interior ofĨ (i) ℓ we can further fix nontrivial functions With this preparation we are finally ready for the main argumentation.
The behavior of the L p -(quasi-)norms in relation to the A 0 p,q -(quasi-)norms is crucial for the proof of Theorem 7.1. Concretely, we have shown for A ∈ {B, F } A s,ᾱ p,q (R d ) = A s,ᾱ p,q (R d ) ⇐⇒ f A 0 p,q (R) ≍ f Lp(R) for band-limited functions f .
On the right-hand side, the (quasi-)norms are thereby all classical and only the univariate case matters. Using known embedding theorems, the exact parameters for equality could therefore be determined (see [49] Section 2.3.2 or [46] Theorem 3.1.1., for example).
Prefering a direct and shorter route, the following lemma provides a simple and quantitative argument for what we need. It investigates the behavior of the respective (quasi-)norms for certain sequences of test functions. As a consequence of statement (i), we extract the necessity p = q = 2 for equality in (49). From (ii) we further obtain p = q in the B-case. Statement (iii) yields 1 < p < ∞ in the F-case. Altogether, this shows that the Sobolev spaces in Theorem 6.1 are precisely those, where equality holds true. 1 , p < 1, N 1/p , p ≥ 1.

Remark 7.2
In case q = ∞ we need to interpret N 1/q ≍ 1. Further, the case A = B with p = ∞ is not considered in Lemma 7.1. By an analogous argument, one can show however that (ii) holds true also for p = ∞. So, for B 0 ∞,q (R) we have the necessary condition q = ∞ to be equivalent to L ∞ (R). It is further not difficult to show that the sequence (f   where (ψ j,k ) j,k shall be a compactly supported, orthogonal, and L ∞ -normalized wavelet system with sufficient vanishing moments and smoothness to characterize the space A 0 p,q (R). Further, for each j ∈ N 0 and k ∈ {0, . . . , 2 j }, we assume the support condition supp (ψ j,k ) ⊂ [0, 1]. Now we note that in a univariate setting, as considered here, we have the coincidencẽ A 0 p,q (R) = A 0 p,q (R). Hence, using the wavelet isomorphism established by Theorem 4.2 for the F-scale and taking into account Remark 4.1 for the B-scale, we immediately obtain Next, we define for j ∈ N 0 the auxiliary functions Observe that 0 ≤ F j ≤ 1. Then, in case 0 < p ≤ 2, In case 2 < p < ∞, we again argue with Hölder Further, since 2/p < 1, Altogether, these estimates show Eε( f N,ε p p ) ≍ N p/2 . As a consequence, we can choose f N := f N,ε(N ) such that f N p p ≍ N p/2 , or equivalently f N p ≍ √ N. ad (ii): With the same wavelet system (ψ j,k ) j,k as before, L ∞ -normalized, define where k(j) is chosen such that the (spatial) support of the wavelets is mutually disjoint. Then, using again the wavelet isomorphism from Theorem 4.2 and Remark 4.1, we deduce ad (iii): Finally, let (ϕ j ) j be a (standard) dyadic resolution of unity, with ϕ 0 = 1 in a neighborhood of 0 and ϕ 1 = ϕ 0 (·/2) − ϕ 0 , and put f N := F −1 ϕ 0 (2 −N ·) .