# Interpolation of Gentle Spaces

Authors: Mourad Ben Slimane; Hnia Ben Braiek
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

The notion of gentle spaces, introduced by Jaffard, describes what would be an “ideal” function space to work with wavelet coefficients. It is based mainly on the separability, the existence of bases, the homogeneity, and theγ-stability. We prove that real and complex interpolation spaces between two gentle spaces are also gentle. This shows the relevance and the stability of this notion. We deduce that Lorentz spaces Lp,q and Hp,q spaces are gentle. Further, an application to nonlinear approximation is presented.


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## Body

## 1. Introduction

Interpolation is a powerful technique for proving continuity of linear operators. Let us recall some basic notions concerning interpolation between Banach spaces. LetX0 and X1 be two Banach spaces. We say that (X0,X1) is a compatible couple if X0 and X1 are continuously embedded in a common Hausdorff topological vector space ℋ (we write Xi↪H for i=0,1).Let(X0,X1) be a compatible couple. Then X0∩X1 and X0+X1 are Banach spaces under the norms
(1)∥f∥X0∩X1=max⁡{∥f∥X0,∥f∥X1},∥f∥X0+X1=inf⁡{∥f0∥X0+∥f1∥X1:f=f0+f1},
where the infimum extends over all representations f=f0+f1 of f with f0∈X0 and f1∈X1.If(X0,X1) is a compatible couple, then a Banach space X is said to be an intermediate space between X0 and X1 if X is continuously embedded between X0∩X1 and X0+X1.Let(Y0,Y1) be a second compatible couple. A linear operator T defined on X0+X1 and taking values in Y0+Y1 is said to be admissible with respect to couples (X0,X1) and (Y0,Y1) if, for each i=0,1, the restriction of T to Xi is a linear continuous operator from Xi into Yi (if Xi=Yi, then T is said to be admissible with respect to (X0,X1)). One looks for intermediate spaces X and Y of the couples (X0,X1) and (Y0,Y1), respectively, such that every admissible operator T maps X into Y. The pair (X,Y) is called interpolation pair relative to (X0,X1) and (Y0,Y1). If Xi=Yi for i=0,1 and X=Y, then X is called an interpolation space between X0 and X1.Note thatX0, X1, X0∩X1, and X0+X1 are examples of interpolation spaces between X0 and X1. Other examples can be constructed by several methods.In 1926, Riesz found the first interpolation method for(Lp(dμ),Lq(dμ)). A generalized version was given by Thorin in 1939/1948 and is known as the Convexity Theorem of Riesz and Thorin or the Riesz-Thorin interpolation theorem. There are many extensions of this theorem. In this connection, we mention the Marcinkiewicz interpolation theorem (in 1939) which extends the Riesz-Thorin interpolation theorem to couples of weak Lp-spaces and which was proved by Zygmund in 1956. In 1958, Stein and Weiss generalized the method for couples (Lp(dμ),Lq(dν)) with different measures μ and ν. At the end of 1958, Lions gave the first proof of the interpolation theorem for quadratic interpolation between Hilbert spaces. Since then several authors have introduced and developed different interpolation methods for couples of general Banach spaces. We mention here essentially two methods: the real interpolation method introduced by Lions and Peetre and the complex method developed by Lions, Calderón, and Krejn. In general, these methods lead to different interpolation spaces. Let us quote the example of usual complex spaces Lp=Lp(ℝd) of p-integrable functions and homogeneous Bessel-potential spaces W˙s,p=W˙s,p(ℝd). If we denote by (X0,X1)[θ] and (X0,X1)θ,q, with 0<θ<1, the interpolation spaces obtained, respectively, by the complex and the real method, then we have (see for instance [1])(i)
(Lp0,Lp1)[θ]=Lp, 1≤p0, p1≤∞ and 1/p=((1-θ)/p0)+(θ/p1).(ii)
(Lp0,Lp1)θ,q=Lp,q, 1≤p0, p1≤∞, 1/p=((1-θ)/p0)+(θ/p1) and p0<q≤∞.(iii)
(W˙s,p0,W˙s,p1)θ,p=W˙s,p, s∈ℝ, 1≤p0,  p1≤∞ and (1/p)=((1-θ)/p0)+(θ/p1).(iv)
(W˙s0,p,W˙s1,p)θ,q=B˙ps,q, s0≠s1, s=(1-θ)s0+θs1 and 1≤p,  q≤∞.(v)
(W˙s0,p0,W˙s1,p1)[θ]=W˙s,p, s0≠s1, s=(1-θ)s0+θs1 and (1/p)=((1-θ)/p0)+(θ/p1).Here,Lp,q=Lp,q(ℝd) and B˙ps,q=B˙ps,q(ℝd) are, respectively, Lorentz and homogenous Besov spaces (see [1–3]). The spaces Lp(ℝd) for 1<p<∞, W˙s,p(ℝd) where s∈ℝ and 1<p<∞, and B˙ps,q(ℝd) with s∈ℝ and 0<p, q≤∞ are gentle spaces (see [4]). Nonetheless, inhomogeneous Besov spaces Bps,q(ℝd) are not gentle because the homogeneity property (the second requirement in Definition 1) is not verified. Recall that
(2)Bps,q(ℝd)=Lp(ℝd)+B˙ps,q(ℝd)ifs<0,1≤p,q≤∞,Bps,q(ℝd)=Lp(ℝd)∩B˙ps,q(ℝd)ifs>0,1≤p,q≤∞.
This means that the interpolation space between two gentle spaces is not always gentle (recall that, above, we said that X0, X1, X0∩X1, and X0+X1 are examples of interpolation spaces between X0 and X1). However, in the third section of this paper, we will prove that gentleness is stable by real and complex interpolation methods. In the next section, we give all the necessary recalls concerning these methods.The notion of gentleness was introduced by Jaffard [5]. It describes what would be an “ideal” function space to work with wavelet coefficients, in “any” wavelet basis. This is the case for Sobolev spaces in PDEs (see [6, 7] for instance), and for Besov spaces in statistics, see [8]. Moreover, many signals and images are stored, denoised, or transmitted by their wavelet coefficients (see [9]). One often needs to obtain local or global information on signals or images by conditions bearing on the moduli of their wavelet coefficients. These conditions should be independent of the chosen wavelet basis.Gentleness is based mainly on separability, existence of bases, homogeneity, andγ-stability; let 𝒮(ℝd) be the Schwartz space of all complex valued rapidly decreasing C∞ functions on ℝd. Set
(3)𝒮∞(ℝd)={f∈𝒮(ℝd):∫ℝd‍xβf(x)dx=0∀β∈ℕd},
(where xβ=x1β1⋯xdβd for all x=(x1,…,xd)∈ℝd and all β=(β1,…,βd)∈ℕd). By 𝒮∞′(ℝd) we denote its topological dual (also called the space of tempered distributions modulo polynomials).Definition 1.
LetH∈ℝ. A function space E is gentle of order H if we have the following.(i)
E is a Banach or a quasi-Banach space of distributions.(ii)
E is homogeneous of order H; that is, there exists a constant C>0 such that for all f∈E, all a∈ℝd and all r>0(4)∥τaf∥E≤C∥f∥E,∥hrf∥E≤CrH∥f∥E,
where τa and hr are the shift and the dilation operators defined by
(5)τaf(x)=f(x-a),hrf(x)=f(rx).(iii)
𝒮∞(ℝd)↪E↪𝒮∞′(ℝd).(iv)
IfE is separable, then 𝒮∞(ℝd) is dense in E, and if E is the dual of a separable space F, then 𝒮∞(ℝd) is dense in F.(v)
There existsγ>0 such that E is γ-stable.The first requirement is explained in the following definition introduced by Bourdaud in [10]. Denote by 𝒟′(ℝd) the dual space of the space of all complex valued compactly supported C∞ functions on ℝd.Definition 2.
A Banach (resp., quasi-Banach) space of distributions is a vector subspaceE of 𝒟′(ℝd) endowed with a complete norm (resp., quasi-norm) such that the embedding E↪𝒟′(ℝd) is continuous.Recall that a quasi-Banach space is a complete topological vector space endowed with a quasi-norm. A quasi-norm (see [1] page 59) satisfies the requirements of a norm except for the triangular inequality which is replaced by the weaker condition
(6)hhhhh∃C>0;∀x,y∈E,∥x+y∥E≤C(∥x∥E+∥y∥E).
We say that ∥·∥E is a p-norm where 0<p≤1 if in addition
(7)∀x,y∈E,∥x+y∥Ep≤∥x∥Ep+∥y∥Ep.
Note that a quasi-norm is always equivalent to a p-norm (see [11]). The real Hardy spaces Hp(ℝd) and Besov spaces B˙ps,p(ℝd), with s∈ℝ and 0<p<1, are quasi-Banach spaces.We should point out that the definitions of intermediate and interpolation spaces carry over without change for a compatible couple of quasi-Banach spaces (see [12–14]).In the second requirement in Definition1, the shift invariance implies that the definition of pointwise regularity introduced by Jaffard in [5] (which describes how the norm of f (properly renormalized by substracting a polynomial) behaves in small neighbourhoods of a given point x0) is the same at every point, and the dilation invariance is implicit in pointwise regularity through scaling invariance. This requirement and the third one imply that Meyer-Lemarié wavelets [15] belong to E since they are obtained by translations and dyadic dilations of a basic function (mother wavelet) in 𝒮∞(ℝd).The fourth requirement shows that wavelet bases are either unconditional bases or unconditional*-weak bases of gentle spaces. Examples of nonseparable spaces for which wavelets are *-weak bases include homogeneous Hölder spaces C˙s(ℝd) and, more generally, homogeneous Besov spaces B˙ps,q(ℝd) with p=∞ or q=∞.The last requirement implies that the characterization of gentle space by wavelet coefficients does not depend on the particularr-smooth wavelet basis (see [15]) which is chosen for γ<r; let ℳγ be the space of infinite matrices M(λ,λ′) indexed by dyadic cubes λ=k2-j+[0,2-j[d and λ′=k′2-j′+[0,2-j′[d (where j,j′∈ℤ and k,k′∈ℤd) and satisfying
(8)∃C>0:∀(λ,λ′)|M(λ,λ′)|≤Cωγ(λ,λ′),
where
(9)ωγ(λ,λ′)=2-((d/2)+γ)|j-j′|(1+(j-j′)2)(1+2inf⁡{j,j′}|k2-j-k′2-j′|)d+γ
(here |·| denotes the Euclidean norm). Meyer proved that ℳγ is algebra. Besides, he defined 𝒪p(ℳγ) as the algebra of bounded operators on L2(ℝd) whose matrices on a r-smooth wavelet basis (for a r>γ) belong to ℳγ and showed that this definition does not depend on the chosen wavelet basis. In particular, we can use compactly supported wavelet bases.Definition 3.
Letγ>0. A Banach or a quasi-Banach space of distributions E is γ-stable if the operators of 𝒪p(ℳγ) are continuous on E.In [16], we extended the notion of gentle spaces to include anisotropic homogeneous Besov spaces.In the fourth section of this paper, we will apply our results to exhibit new examples of gentle spaces, namely, Lorentz spacesLp,q (see [2]) and Hp,q spaces (see [17]). We will also prove that if the Jackson and Bernstein inequalities are valid, then “nonlinear approximation space” as defined in [18] associated to a gentle space is gentle. Note that there are different types of nonlinear approximations. The n-term approximation is one of the dominant types. We can mention, for example, approximation by splines with n free knots or by rational functions of degree n; see DeVore and Popov [19]. Approximation by a linear combination with n-term of φ-function was developed by DeVore et al. in [20]. A generalization of n-term approximation (called restricted approximation) by a linear combination of compactly supported biorthogonal wavelets was presented by Cohen et al. in [21]. In this paper, we consider approximation by a linear combination of Lemarié-Meyer wavelets as was done by Kyriazis in [18]. This form of approximation occurs in several applications including image processing, statistical estimation, and numerical solutions of differential equations.


## 2. Real and Complex Interpolation Methods

Originally, real and complex interpolation methods were developed for Banach spaces. The extension of real interpolation for quasi-Banach spaces causes no serious problem (see [12, 13, 22]). However, for the complex method the situation is quite different (see [23–25]). Let us recall briefly some basic definitions and notations related to these two methods. For more details see [1, 12, 13, 22, 26–28].Definition 4.
Let(X0,X1) be a compatible couple of Banach or quasi-Banach spaces. (1)
TheK-functional is defined for each f∈X0+X1 and t>0 by
(10)K(f,t)=K(f,t,X0,X1)=inf⁡{∥f0∥X0+t∥f1∥X1:f=f0+f1},
where the infimum extends over all representations f=f0+f1 of f with f0∈X0 and f1∈X1.(2)
TheJ-functional is defined for each f∈X0∩X1 and t>0 by
(11)J(f,t)=J(f,t,X0,X1)=max⁡{∥f∥X0,t∥f∥X1}.TheK-functional and J-functional, introduced By Peetre, are nonnegative concave and increasing functions.Definition 5.
Let(X0,X1) be a compatible couple of Banach or quasi-Banach spaces. (1)
Let0<θ<1 and 0<q<∞ or let 0≤θ≤1 and q=∞. The space (X0,X1)θ,q,K consists of all f∈X0+X1 such that
(12)∥f∥θ,q,K:={(∫0∞‍(t-θK(f,t))qdtt)1/q,0<θ<1,0<q<∞,sup⁡t>0⁡t-θK(f,t),0≤θ≤1,q=∞
is finite.(2)
Let0<θ<1 and 0<q≤∞. The space (X0,X1)θ,q,J consists of all f∈X0+X1 that are represented by Bochner-integral
(13)f=∫0∞‍u(s)dss,
where u is measurable with values in X0∩X1 and such that
(14)∥f∥θ,q,J:=inf⁡{(∫0∞‍(s-θJ(u(s),s))qdss)1/q,q<∞,sup⁡s>0⁡s-θJ(u(s),s),q=∞
is finite, where the infimum is taken over all representations (13) of f.Remark 6.
There is a discrete representation of the space(X0,X1)θ,q,J (see [1]); in fact f∈(X0,X1)θ,q,J if and only if there exists a sequence (uν)ν∈ℤ in X0∩X1 such that
(15)f=∑ν∈ℤ‍uν,(16)(∑ν∈ℤ‍(2-νθJ(uν,2ν))q)1/q<∞,ifq<∞
or
(17)sup⁡ν∈ℤ⁡2-νθJ(uν,2ν)<∞,ifq=∞.
Moreover,
(18)∥f∥θ,q,Jisequivalenttoinf⁡{(∑ν∈ℤ‍(2-νθJ(uν,2ν))q)1/q,q<∞,sup⁡ν∈ℤ⁡2-νθJ(uν,2ν),q=∞,
where the infimum extends over all sequences (uν)ν∈ℤ satisfying (15).The following result is given in [1] (see also [29]).Theorem 7.
Let(X0,X1) be a compatible couple of Banach (resp., quasi-Banach) spaces. Then, spaces (X0,X1)θ,q,K, with 0<θ<1 and 0<q<∞ or 0≤θ≤1 and q=∞, and (X0,X1)θ,q,J, with 0<θ<1 and 0<q≤∞, equipped, respectively, by the norms (resp., quasi-norms) (12) and (14) are Banach (resp., quasi-Banach) spaces and are interpolation spaces between X0 and X1.
Furthermore, if0<θ<1 and 0<q≤∞, then (X0,X1)θ,q,K=(X0,X1)θ,q,J with equivalence of norms (resp., quasi-norms).Real interpolation method means either theK- or J-method. We will write (X0,X1)θ,q instead of (X0,X1)θ,q,K or (X0,X1)θ,q,J, if 0<θ<1. If θ=0 or 1 and q=∞, then (X0,X1)θ,q denotes (X0,X1)θ,q,K. By ∥·∥θ,q we denote the norm or quasi-norm on (X0,X1)θ,q depending whether this space is Banach or quasi-Banach.In the complex case, we will first restrict ourselves to Banach spaces. There are two interpolation spaces whose norms are equivalent under some conditions. Set(19)S={z∈ℂ,0≤ℜ(z)≤1},S0={z∈ℂ,0<ℜ(z)<1}.
Let (X0,X1) be a compatible couple of Banach spaces. We denote by ℱ the space of all functions f:S→X0+X1 that are bounded, continuous on S, and analytic on S0, such that functions t↦f(k+it), k=0,1, from ℝ into Xk, are continuous and tend to zero as |t|→∞.By𝒢 we denote the space of all functions g:S→X0+X1 that are continuous on S and analytic on S0, satisfying
(20)∥g(z)∥X0+X1≤C(1+|z|)
such that g(k+it1)-g(k+it2) has values in Xk, k=0,1, for any -∞<t1<t2<∞.Spacesℱ and 𝒢 provided, respectively, with
(21)∥f∥ℱ:=max⁡{sup⁡t∈ℝ∥f(it)∥X0,sup⁡t∈ℝ∥f(1+it)∥X1},(22)∥g∥𝒢:=max⁡{sup⁡t1,t2∈ℝ∥g(it1)-g(it2)t1-t2∥X0,hhhhhhhhhhhhsup⁡t1,t2∈ℝ∥g(1+it1)-g(1+it2)t1-t2∥X1}
are Banach spaces.Definition 8.
Let(X0,X1) be a compatible couple of Banach spaces. For all 0≤θ≤1 (resp. 0<θ<1) we define (X0,X1)[θ] (resp., (X0,X1)[θ]) as the space of all x∈X0+X1 such that x=f(θ) (resp., x=g′(θ)) for some f∈ℱ (resp., g∈𝒢).Theorem 9.
Let(X0,X1) be a compatible couple of Banach spaces. Then, spaces (X0,X1)[θ], where 0≤θ≤1 and (X0,X1)[θ], where 0<θ<1, equipped respectively with
(23)∥x∥[θ]:=inf⁡{∥f∥ℱ:f(θ)=x,f∈ℱ},(24)∥x∥[θ]:=inf⁡{∥g∥𝒢:g′(θ)=x,g∈𝒢}
are Banach spaces and are interpolation spaces between X0 and X1.Remark 10.
We have(X0,X1)[θ]⊂(X0,X1)[θ]. In general, (X0,X1)[θ] and (X0,X1)[θ] are not equal. However, if either X0 or X1 is reflexive and if 0<θ<1, then (X0,X1)[θ]=(X0,X1)[θ] and ∥x∥[θ]=∥x∥[θ], for all x∈X0+X1.The extension of this method to quasi-Banach spaces is not routine; one cannot use duality as was done for Banach spaces. The duality theorem is not true in general in the quasi-Banach case, and the maximum principle fails for functions taking values in a quasi-Banach space (see [30]).There are several possible ways to define complex interpolation spaces. For example, in [23] complex interpolation was defined in the framework of Fourier analysis, while in [24] complex interpolation was defined as in the Banach setting but by adding in ∥f∥ℱ (which was given in (21)) a third term sup⁡z∈S0∥f(z)∥X0+X1. In [25], the authors described a new approach to interpolate by the complex method some quasi-Banach spaces; let A(S) be the space of all scalar valued functions f continuous and bounded on S and analytic on S0. Let (X0,X1) be a compatible couple of quasi-Banach spaces. Denote by 𝒜(X0,X1) the collection of all functions f that can be written as a finite sum f(z)=∑k‍fk(z)ak where fk∈A(S) and ak∈X0∩X1. We put
(25)∥f∥𝒜(X0,X1):=max⁡{sup⁡t∈ℝ∥f(it)∥X0,sup⁡t∈ℝ∥f(1+it)∥X1},
and for all x∈X0∩X1 and 0<θ<1 let
(26)∥x∥θ:=inf⁡{∥f∥𝒜(X0,X1):f(θ)=x,f∈𝒜(X0,X1)}.
This functional is a semi-quasi-norm. Let
(27)N={x∈X0∩X1:∥x∥θ=0};
then, ((X0∩X1)/N,∥·∥θ) is a quasi-normed space. We define [X0,X1]θ as the completion of ((X0∩X1)/N,∥·∥θ) (see [31]).


## 3. Real and Complex Interpolation between Gentle Spaces

Assume thatE0 and E1 are two gentle spaces. Let E be an interpolation space between them. Then, E is a Banach or a quasi-Banach space of distributions and 𝒮∞(ℝd)↪E↪𝒮∞′(ℝd).On the other hand, there exist two constantsγ0>0 and γ1>0 such that Ei is γi-stable, i=0,1. Set γ=min⁡{γ0,γ1}. If T∈𝒪p(ℳγ), then T∈𝒪p(ℳγi). So T is continuous on Ei and therefore continuous on E. Thus, E is γ-stable.We will now prove the homogeneity property.Proposition 11.
LetE0 and E1 be two gentle spaces of order H0 and H1, respectively. Then, spaces (1)
(E0,E1)θ,q, with0<θ<1and1≤q<∞or0≤θ≤1andq=∞,(2)
(E0,E1)[θ],with0≤θ≤1(in the Banach setting),(3)
(E0,E1)[θ],with0<θ<1(in the Banach setting),(4)
[E0,E1]θ,with0<θ<1(in the quasi-Banach setting) are homogeneous spaces of order (1-θ)H0+θH1.Proof.
(1) Let0<θ<1 and 0<q<∞ or let 0≤θ≤1 and q=∞. Let f∈(E0,E1)θ,q, f0∈E0, and f1∈E1 such that f=f0+f1. Let t>0. From the homogeneity of E0 and E1 we get
(28)K(τa(f),t)≤∥τa(f0)∥E0+t∥τa(f1)∥E1≤C(∥f0∥E0+t∥f1∥E1),K(hr(f),t)≤∥hr(f0)∥E0+t∥hr(f1)∥E1≤CrH0(∥f0∥E0+trH1-H0∥f1∥E1).
By taking the infimum over all such decompositionsf=f0+f1 of f, we obtain
(29)K(τa(f),t)≤CK(f,t)so∥τa(f)∥θ,q≤C∥f∥θ,q,K(hr(f),t)≤CrH0K(f,trH1-H0).(i)
If0<θ<1 and 0<q<∞, then
(30)∥hr(f)∥θ,q=(∫0∞‍(t-θK(hr(f),t))qdtt)1/q≤CrH0(∫0∞‍(t-θK(f,trH1-H0))qdtt)1/q.
It follows, by a simple change of variable, that(31)∥hr(f)∥θ,q≤CrH0r-θ(H0-H1)(∫0∞‍(t-θK(f,t))qdtt)1/q=Cr(1-θ)H0+θH1∥f∥θ,q.(i)
If0≤θ≤1 and q=∞, we obtain
(32)∥hr(f)∥θ,q=sup⁡t>0⁡t-θK(hr(f),t)≤Cr(1-θ)H0+θH1∥f∥θ,q.
Therefore, the space(E0,E1)θ,q is homogeneous of order (1-θ)H0+θH1.
(
2
) Now, let f∈(E0,E1)[θ]; there exists g∈ℱ such that f=g(θ). Clearly
(33)τa(g(z))(x)=g(z)(x-a),hr(g(z))(x)=g(z)(rx).
Thus,z↦τa(g(z)) and z↦hr(g(z)) are analytic functions.
From (23), (21), and the homogeneity of E0 and E1, we can easily see that
(34)∥τa(f)∥[θ]≤C∥f∥[θ].
Put(35)F(z)=r(z-1)H0r-zH1hr(g(z)).
Therefore,(36)hr(f)=r(1-θ)H0+θH1F(θ).
Hence,(37)∥hr(f)∥[θ]≤r(1-θ)H0+θH1∥F∥ℱ=r(1-θ)H0+θH1max⁡{sup⁡t∈ℝ∥F(it)∥E0,hhhhhhhhhhhhhhhhsup⁡t∈ℝ∥F(1+it)∥E1}.
It follows from (35) and the homogeneity of E0 and E1 that
(38)∥F(it)∥E0=r-H0∥hr(g(it))∥E0≤C∥g(it)∥E0,∥F(1+it)∥E1=r-H1∥hr(g(1+it))∥E1≤C∥g(1+it)∥E1.
This implies that
(39)∥F∥ℱ≤C∥g∥ℱ,∥hr(f)∥[θ]≤Cr(1-θ)H0+θH1∥g∥ℱ.
Taking the infimum over all g∈ℱ such that g(θ)=f, we obtain
(40)∥hr(f)∥[θ]≤Cr(1-θ)H0+θH1∥f∥[θ].
(
3
) If f∈(E0,E1)[θ], then there exists g∈𝒢 such that f=g′(θ). So as previously z→τa(g(z)) and z→hr(g(z)) are analytic functions. Clearly, from (24), (22), and the homogeneity of E0 and E1, we have
(41)∥τa(f)∥[θ]≤C∥f∥[θ].
Set
(42)G(z)=∫0z‍r(η-1)H0r-ηH1hr(g′(η))dη.
Hence,
(43)G′(θ)=r(θ-1)H0r-θH1hr(f).
Thus,
(44)∥hr(f)∥[θ]≤r(1-θ)H0+θH1∥G∥𝒢.
Let -∞<t1<t2<∞. From (42) and the fact that E0 is homogeneous of order H0, we get
(45)∥G(it1)-G(it2)∥E0≤r-H0∫t1t2‍∥hr(g′(it))∥E0dt≤C∫t1t2‍∥g′(it)∥E0dt≤C∥g(it2)-g(it1)∥E0.
Similarly
(46)∥G(1+it1)-G(1+it2)∥E1≤C∥g(1+it2)-g(1+it1)∥E1.
Therefore,
(47)∥G∥𝒢≤C∥g∥𝒢,so∥hr(f)∥[θ]≤Cr(1-θ)H0+θH1∥g∥𝒢.
By taking the infimum over all g∈𝒢 such that g′(θ)=f, we obtain
(48)∥hr(f)∥[θ]≤Cr(1-θ)H0+θH1∥f∥[θ].
(
4
) Now, let f∈[E0,E1]θ. There exists a sequence (fn)n∈(E0∩E1)/N such that ∥f-fn∥θ→0. Let g∈𝒜(E0,E1) such that g(θ)=f-fn and g is written as a finite sum g(z)=∑k‍gk(z)ak where gk∈A(S) and ak∈E0∩E1. We define
(49)τa(g(z))=∑k‍τa(gk(z))ak,hr(g(z))=∑k‍hr(gk(z))ak,
where
(50)τa(gk(z))(x)=gk(z)(x-a),hr(gk(z))(x)=gk(z)(rx),
and z↦τa(gk(z)) and z↦hr(gk(z)) are analytic functions. By arguing similarly as in (E0,E1)[θ], by taking f-fn instead of f, we get the desired result.Let us now prove the density property.Proposition 12.
Ifq<∞, then 𝒮∞(ℝd) is dense in (E0,E1)θ,q.Proof.
Sinceq<∞, then E0∩E1 is dense in (E0,E1)θ,q (see e.g., [1]). Therefore it suffices to prove that every function f∈E0∩E1 can be approached by a sequence (fn)n∈ℕ∈𝒮∞(ℝd) in (E0,E1)θ,q. The assumption q<∞ implies that 0<θ<1, so by the equivalence between real interpolation spaces we will prove this approach in (E0,E1)θ,q,J.
On the other hand, sinceE0 and E1 are gentle spaces, then wavelets are either unconditional bases or unconditional *-weak bases, depending on the separability of these spaces. Using the γ-stability, we can take Lemarié-Meyer wavelet basis; let ψ0 and ψ1 be, respectively, the Lemarié-Meyer [32] father and mother wavelets; that is, ψ0 and ψ1 are in the Schwartz class such that all moments of ψ1 vanish, ∫ℝ‍ψ0(x)dx=1, and the collection of the union of (ψ0(·-k))k∈ℤ and (2j/2ψ1(2j·-k))j∈ℕ,k∈ℤ is an orthonormal basis of L2(ℝ). Let V be the set of nonzero vertices of the unit cube in ℝd, for each vertex v=(v1,…,vd)∈V we set
(51)ψv(x):=ψv1(x1)⋯ψvd(xd),x=(x1,…,xd)∈ℝd,
and we define
(52)ψλv(x):=2dj/2ψv(2jx-k),
indexed by dyadic cube λ=2-j[k,k+1], where k∈ℤd and j∈ℤ. Denote by Λ (resp., Λj) the set of all dyadic cubes (resp., dyadic cubes at scale j). Then, the collection
(53){ψλv:λ∈Λ,v∈V}
is an orthonormal basis of L2(ℝd). Thus any function f∈L2(ℝd) can be written as
(54)f=∑λ∈Λ‍∑v∈V‍cλvψλv,wherecλv=〈f,ψλv〉.
Let f∈E0∩E1. Put fj=∑λ∈Λj‍∑v∈V‍cλvψλv; then, (fj)j∈ℤ belongs to 𝒮∞(ℝd), so to E0∩E1 and f=∑j∈ℤ‍fj. Since E0∩E1⊂(E0,E1)θ,q,J, we therefore obtain from (15) and (16) that
(55)(∑j∈ℤ‍(2-jθJ(fj,2j))q)1/q<∞.
Set now
(56)fn=∑j=-nn‍fj.
It follows from (18) that
(57)∥f-fn∥θ,q,J≤(∑|j|>n‍(2-jθJ(fj,2j))q)1/q,
which tends to 0 as n tends to infinity.Proposition 13.
Suppose that𝒮∞(ℝd) is dense in E0 and in E1. (1)
Assume thatE0andE1are Banach spaces. Then
(a)
If0≤θ≤1, then𝒮∞(ℝd)is dense in(E0,E1)[θ].(b)
If0<θ<1, then𝒮∞(ℝd)dense in(E0,E1)[θ].(2)
Assume thatE0andE1are quasi-Banach spaces. Then𝒮∞(ℝd)is dense in[E0,E1]θ, where0<θ<1.Proof .
Letf∈(E0,E1)[θ]. There exists g∈ℱ such that f=g(θ).
Since𝒮∞(ℝd) is dense in both E0 and E1, then, for all ε>0, there exist two functions g0 and g1 in 𝒮∞(ℝd) such that for all t∈ℝ,
(58)∥g(it)-g0∥E0≤ε,∥g(1+it)-g1∥E1≤ε.
Let H∈ℱ such that H takes values in 𝒮∞(ℝd), H(it)=g0, and H(1+it)=g1 for all t∈ℝ. Put h=H(θ). Thus, from (23), (21), and (58), we obtain
(59)∥f-h∥[θ]≤∥g-H∥ℱ≤ε.
Since h∈𝒮∞(ℝd), then this implies the density of 𝒮∞(ℝd) in (E0,E1)[θ].
Now letf∈(E0,E1)[θ]. Arguing similarly as previously but by replacing ℱ by 𝒢, g by g′, and (58) by
(60)∥g(it1)-g(it2)t1-t2-g0∥E0≤ε,∥g(1+it1)-g(1+it2)t1-t2-g1∥E1≤ε.
Taking H∈𝒢 such that H takes values in 𝒮∞(ℝd), H(it)=tg0 and H(1+it)=tg1 for all t∈ℝ. To finish the proof, it suffices to put h=H′(θ).
For the proof of the density of𝒮∞(ℝd) in [E0,E1]θ, we argue similarly as in (E0,E1)[θ] by taking f-fn instead of f, where (fn)n is a sequence in (E0∩E1)/N such that ∥f-fn∥θ→0.Consequence. Suppose that 𝒮∞(ℝd) is dense in both X0 and X1, where Xi′=Ei, with i=0,1. Let 0<θ<1. It follows from the previous proposition that 𝒮∞(ℝd) is dense in both (X0,X1)[θ] and (X0,X1)[θ]. On the other hand, since
(61)𝒮∞(ℝd)↪X0∩X1↪Xi,i=0,1,
then X0∩X1 is dense in Xi, i=0,1. From the duality theorem (see [1]), it follows that
(62)(X0,X1)[θ]′=(X0′,X1′)[θ]=(E0,E1)[θ],((X0,X1)[θ])′=(X0,X1)[θ]′.
Furthermore, if either E0 or E1 is reflexive (then either X0 or X1 is reflexive), then
(63)(X0,X1)[θ]′=(E0,E1)[θ],((X0,X1)[θ])′=(X0′,X1′)[θ]=(E0,E1)[θ].


## 4. Applications

As a first application we show that Lorentz spacesLp,q=Lp,q(ℝd) are gentle. This follows from the fact that Lp,q=(Lp0,Lp1)θ,q (see [1]), where 0<θ<1, 1<p0,p1<∞, (1/p)=((1-θ)/p0)+(θ/p1) and p0<q≤∞ and Lp0=Lp0(ℝd) and Lp1=Lp1(ℝd) (where 1<p0,p1<∞) are gentle spaces (see [4]).In the second application, we consider the real interpolation between Hardy spacesHp0=Hp0(ℝd) and BMO=BMO(ℝd) spaces. In [4], we investigated the gentleness property of some functional spaces and we proved the stability of this property by duality. We also proved that homogeneous Lizorkin-Triebel spaces F˙ps,q(ℝd) (see [33]), where s∈ℝ, 0<p<∞ and 0<q≤∞ are gentle spaces. In particular, Hp(ℝd)=F˙p0,2(ℝd), where 0<p≤1 are gentle. As a consequence, since BMO is the topological dual of H1, then BMO is gentle. In [34], it was shown that, for 0<θ<1, 0<q≤∞, 0<p0<1, and p=p0/1-θ, (Hp0,BMO)θ,q=Hp,q, where Hp,q=Hp,q(ℝd) is the space of all tempered distributions f on ℝd such that
(64)sup⁡t>0t-d|ϕt*f|∈Lp,q,
where ϕt(x)=ϕ(x/t) and ϕ is a smooth function with ∫ℝd‍ϕ(x)dx≠0 (see [17]). Consequently Hp,q are gentle spaces. Note that Hp,q=Hp for p=q.Our last application refers to some nonlinear approximation spaces introduced in [18]. Take the Lemarié-Meyer orthonormal basis (53). Let Σn be the set of all functions
(65)S=∑λ∈Λ(n)‍Aλ(S),
where Aλ(S)=∑v∈V‍cλvψλv and Λ(n) is a subset of Λ with cardinality ♯Λ(n)≤n. For a given distribution f and any quasi-normed subspace X⊂𝒮∞′(ℝd), we define
(66)σn(f,X):=inf⁡S∈Σn∥f-S∥X.
For 0<q≤∞ and α>0, we define the approximation class 𝒜qα(X) as the set of all f∈𝒮∞′(ℝd) such that
(67)|f|𝒜qα(X)={(∑j∈ℤ‍(2jασ2j(f,X))q)1/q,0<q<∞,sup⁡j∈ℤ2jασ2j(f,X),q=∞,
is finite.In [18], Kyriazis proved that if X and Y are quasi-normed spaces that verify, for some r>0, Jackson inequality (forallf∈Y,σn(f,X)≤Cn-r/d∥f∥Y) and Bernstein inequality (forallS∈Σn,∥S∥Y≤Cnr/d∥S∥X), then for each 0<α<r and 0<q≤∞(68)𝒜qα/d(X)=(X,Y)α/r,q.
This result was also proved in [21], where Cohen et al. have used compactly supported biorthogonal wavelets, and in [35], by DeVore and Lorentz, where Y is embedded in X.In the literature complete characterizations are known for the casesX=Hp, 0<p<∞; X=B˙p0,p, 0<p<∞; and X=F˙pβ,t, β∈ℝ, 0<p<∞, and 0<t≤∞. DeVore et al. have shown, in [20], that for any 0<α<s and any 0<q≤∞(69)𝒜qα/d(Hp)=(Hp,B˙τs,τ)α/s,q,1τ=sd+1p.
This result was proved by simpler techniques in [21], where the authors proved also that for any 0<α<s and any 0<q≤∞(70)𝒜qα/d(B˙p0,p)=(B˙p0,p,B˙τs,τ)α/s,q
and for any α>0 and any 0<q≤∞(71)𝒜qα/d(Hp)=𝒜qα/d(B˙p0,p).
Approximation in F˙pβ,t was investigated in [18]. More precisely, it has been shown that, for every 0<p<∞, 0<q,  t≤∞, α>0, and β∈ℝ,
(72)𝒜qα/d(F˙pβ,t)=(F˙pβ,t,B˙τγ,τ)α/(γ-β),q,
where α<γ-β and 1/τ=γ-β/d+1/p.SinceHp, B˙p0,p, and F˙pβ,t are gentle spaces, then their approximation spaces (which are real interpolation spaces) are also gentle. Moreover, the pairs (Hp,B˙τs,τ), (B˙p0,p,B˙τs,τ), and (F˙pβ,t,B˙τγ,τ), under the above conditions on all parameters p, q, s, t, τ, and γ, verify the Jackson and Bernstein inequalities. More generally, we deduce that if X and Y are gentle spaces and if the Jackson and Bernstein inequalities are valid, then the approximation space is also gentle.In [18, 21], the authors restricted themselves to particular wavelets. They remarked that all of their theorems hold in more generality. Using our result, this remark is now confirmed using the gentleness stability by real interpolation.



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