Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation
aa r X i v : . [ m a t h . C A ] A ug RESTRICTED NON-LINEAR APPROXIMATION INSEQUENCE SPACES AND APPLICATIONS TOWAVELET BASES AND INTERPOLATION
EUGENIO HERN ´ANDEZ AND DANIEL VERA
Abstract.
Restricted non-linear approximation is a type of N-term approximationwhere a measure ν on the index set (rather than the counting measure) is used tocontrol the number of terms in the approximation. We show that embeddings forrestricted non-linear approximation spaces in terms of weighted Lorentz sequencespaces are equivalent to Jackson and Bernstein type inequalities, and also to theupper and lower Temlyakov property. As applications we obtain results for waveletbases in Triebel-Lizorkin spaces by showing the Temlyakow property in this set-ting. Moreover, new interpolation results for Triebel-Lizorkin and Besov spaces areobtained. Introduction
Thresholding of wavelet coefficients is a technique used in image processing to com-press signals or reduce noise. The simplest thresholding algorithm T ε ( ε >
0) of a signal f is obtained by eliminating from a representation of f the terms whose coefficientshave absolute value smaller than ε .Although the thresholding approximants T ε ( f ) are sometimes a visually faithfulrepresentation of f , they are not exact, and from a theoretical point of view an erroris introduced if f is replaced by T ε ( f ). Such errors have initially been measured inthe L − norm, but it is argued in [19] that procedures having small error in L p , oras stated in the statistical community, small L p − risk, may reflect better the visualproperties of a signal. Observe that in the usual thresholding the error is measuredin the same space as the signal is represented, usually L .A more general situation is considered in [6] where the wavelet coefficients arethresholded from a representation of the signal in the Hardy space H r , 0 < r < ∞ (recall that H r = L r if 1 < r < ∞ ), but the error is measured in the Hardy space H p , 0 < p < ∞ . They show that this situation is equivalent to a type of nonlinearapproximation, called restricted , in which a measure ν on the index set of dyadiccubes of R d is used to control the number of terms in the approximation. In theclassical n − term approximation ν ( Q ) = 1, Q ∈ D (counting measure), and in [6] ν ( Q ) = | Q | − p/r .The article [6] provides a description of the approximation spaces in this settingin terms of certain type of discrete Lorentz spaces, as well as interpolation resultsfor certain pairs of H p and Besov spaces. One of the novelties of this article is that, Date : November 21, 2018.2000
Mathematics Subject Classification.
Key words and phrases.
Democracy functions, interpolation spaces, Lorentz spaces, non-linearapproximation, Besov spaces, Triebel-Lizorkin spaces.Research supported by GrantS MTM2007-60952 and MTM2010-16518 of Spain. although the error is measured in H p , the approximation spaces are not necessarilycontained in H p .The theory of restricted nonlinear approximation was further developed in [20] con-sidering the case of a quasi-Banach space X , an unconditional basis B = { e I } I ∈D , and ameasure ν on the countable set D . They show that, in this abstract setting, restrictedthresholding and restricted nonlinear approximation are linked to the p − Temlyakovproperty for ν (see definition in [20]). They also show that this property is equiv-alent to certain Jackson and Bernstein type inequalities and to have the restrictedapproximation spaces identified as discrete Lorentz spaces. The approach in [20] isthat the approximation spaces are contained in X and, hence, not all results in [6] canbe recovered.Denote by S the space of all sequences s = { s I } I ∈D of complex numbers indexed by acountable set D . In the present paper we study restricted nonlinear approximation forquasi-Banach lattices f ⊂ S (see definition in section 2.1). Given a positive measure ν on D we define the restricted approximation spaces A ξµ ( f, ν ), 0 < ξ < ∞ , 0 < µ ≤ ∞ ,as subsets of S using ν to control the number of terms in the approximation and f tomeasure the error (see section 2.2).Denote by E = { e I } I ∈D the canonical basis for S . We use a weight sequence u = { u I } I ∈D , u I >
0, to control the weight of each e I . Discrete Lorentz spaces ℓ µη ( ν )are defined as sequences s = { s I } I ∈D ∈ S using the ν distribution function of thesequence { u I s I } I ∈D (see section 2.5). Here, η is a function in W (see section 2.4) moregeneral than η ( t ) = t /p , 0 < p < ∞ .It is shown in subsections 2.6 and 2.7 that the condition C η ( ν (Γ)) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X I ∈ Γ e I u I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ≤ C η ( ν (Γ)) (1.0.1)for all Γ ⊂ D , with ν (Γ) < ∞ , η ∈ W and η ∈ W + , is equivalent to inclusions be-tween A ξµ ( f, ν ) and ℓ µt ξ η ( t ) ( u , ν ), and also to some Jackson and Bernstein type inequal-ities. When η ( t ) = η ( t ) = t /p , condition (1.0.1) is called in [20] the p − Temlyakovproperty.Working with sequence spaces is not a restriction. Lebesgue, Sobolev, Hardy andLipschitz spaces all have a sequence space counterpart when using the ϕ − transform([10], [11]) or wavelets ([23], [26], [7], [16], [24], [1], [22]). More generally, the Triebel-Lizorkin, f sp,r , and Besov, b sp,r , spaces of sequences (see section 2.9) allow faithful rep-resentations of Triebel-Lizorkin, F sp,r ( R d ), and Besov, B sp,r ( R d ), spaces (these includeall the above spaces). When our results are coupled with the abstract transferenceframework designed in [13] we recover results for distribution or function spaces, asthe case may be. One reason to consider such general setting, besides the obviousgeneralizations, is that measuring the error k f − T ε ( f ) k in Sobolev spaces, where thesmoothing properties of f − T ε ( f ) are taken into account, may give a visually morefaithful representation of f , than when measured in L p . Observe that two functionsmay visually be very different although they may be close in the L p norm.In subsection 2.10 we show that (1.0.1) holds when f = f s p ,q and u I = k e I k f s p ,q ,with η ( t ) = η ( t ) = t /p and ν ( I ) = | I | α if and only if α = p ( s − s d − p ) + 1 = 1 or if α = 1 then p = q . When the results of subsection 2.6 and 2.7 are applied to this case,we show that restricted approximation spaces of Triebel -Lizorkin spaces are identified ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 3 with discrete Lorentz spaces, which coincide with Besov spaces for some particularvalues of the parameters (Lemma 2.10.4). The results in [6] and [18] are simplecorollaries. We also give a result about interpolation of Triebel-Lizorkin and Besovspaces (section 2.11) with less restrictions on the parameters than those considered in[6].The organization of the this paper is as follows. Notation, definitions, results andcomments are given in section 2 which is divided in subsections 2 .x with 1 ≤ x ≤ .x , its proof can be found in subsection 3 .x . Be aware that if asubsection 2 .x only contains notation, definitions and/or comments, but no statementsof results, the corresponding subsection 3 .x does not appear in section 3.2. Notations, Definitions, Statements of Results and Comments
Sequence Spaces.
Let D be a countable (index) set whose elements will bedenoted by I . The set D could be N , Z , ... or, as in the applications we have in mind,the countable set of dyadic cubes on R d .Denote by S = C D the set of all sequences of complex numbers s = { s I } I ∈D definedover the countable set D . For each I ∈ D , we denote by e I the element of S withentry 1 at I and 0 otherwise. We write E = { e I } I ∈D for the canonical basis of S . Weshall use the notation P I ∈ Γ s I e I , Γ ⊂ D , to denote the element of S whose entry is s I when I ∈ Γ and 0 otherwise. Notice that no meaning of convergence is attached tothe above notation even when Γ is not finite.
Definition 2.1.1.
A linear space of sequences f ⊂ S is a quasi-Banach (se-quence) lattice if there is a quasi-norm k·k f in f with respect to which f is completeand satisfies:(a) Monotonicity: if t ∈ f and | s I | ≤ | t I | for all I ∈ D , then s ∈ f and k{ s I }k f ≤k{ t I }k f .(b) If s ∈ f , then lim n →∞ k s I n e I n k f = 0 , for some enumeration I = { I , I , . . . } . We will say that a quasi-Banach (sequence) lattice f is embedded in S , and write f ֒ → S if lim n →∞ k s n − s k f = 0 ⇒ lim n →∞ s ( n ) I = s I ∀ I ∈ D . (2.1.1) Remark 2.1.2.
When E = { e I } I ∈D is a Schauder basis for f , condition (a) in Defi-nition 2.1.1 implies that E is an unconditional basis for f with constant C = 1 . Restricted Non-linear Approximation in Sequence Spaces.
In this paper ν will denote a positive measure on the discrete set D such that ν ( I ) > I ∈ D .In the classical N -term approximation ν is the counting measure (i.e. ν ( I ) = 1 for all I ∈ D ), but more general measures are used in the restricted non-linear approximationcase. The measure ν will be used to control the number of terms in the approximation. Definition 2.2.1.
We say that ( f, ν ) is a standard scheme (for restricted nonlinear approximation) ifi) f is a quasi-Banach (sequence) lattice embedded in S .ii) ν is a measure on D as explained in the first paragraph in this section. EUGENIO HERN ´ANDEZ AND DANIEL VERA
Let ( f, ν ) be a standard scheme. For t >
0, defineΣ t,ν = { t = X I ∈ Γ t I e I : ν (Γ) ≤ t } . Notice that Σ t,ν is not linear, but Σ t,ν + Σ t,ν ⊂ Σ t,ν .Given s ∈ S , the f -error (or f -risk) of approximation to s by elements of Σ t,ν isgiven by σ ν ( t, s ) = σ ν ( t, s ) f := inf t ∈ Σ t,ν k s − t k f . Notice that elements s ∈ S not in f could have finite f -risk since elements of Σ t,ν could have infinite number of entries. Definition 2.2.2. ( Restricted Approximation Spaces ) Let ( f, ν ) be a standardscheme.i) For < ξ < ∞ and < µ < ∞ , A ξµ ( f, ν ) is defined as the set of all s ∈ S suchthat k s k A ξµ ( f,ν ) := (cid:18)Z ∞ [ t ξ σ ν ( t, s )] µ dtt (cid:19) /µ < ∞ . (2.2.1) ii) For < ξ < ∞ and µ = ∞ , A ξ ∞ ( f, ν ) is defined as the set of all s ∈ S such that k s k A ξ ∞ ( f,ν ) := sup t> t ξ σ ν ( t, s ) < ∞ . (2.2.2)Notice that the spaces A ξµ ( f, ν ) depend on the canonical basis E of S . When f areunderstood, we will write A ξµ ( ν ) instead of A ξµ ( f, ν ). Remark 2.2.3. If s ∈ f , using σ ν ( t, s ) ≤ k s k f , it is easy to see that (2.2.1) can bereplaced by k s k f plus the same integral from to ∞ . We need to consider the wholerange < t < ∞ since we do not assume s ∈ f . Similar remark holds for µ = ∞ in(2.2.2). Nevertheless, the properties of the restricted non-linear approximation spacesare the same as the N -term approximation spaces (see [27] or [8] ). By splitting the integral in dyadic pieces and using the monotonicity of the f -error σ ν we have an equivalent quasi-norm for the restricted approximation spaces: k s k A ξµ ( ν ) ≈ ∞ X k = −∞ [2 kξ σ ν (2 k , s )] µ ! /µ . (2.2.3)2.3. The Jackson and Bernstein type inequalities.
It is well known the fun-damental role played by the Jackson and Bernstein type inequalities in non-linearapproximation theory. Considering our standard scheme ( f, ν ) we give the followingdefinitions.
Definition 2.3.1.
Given r > , a quasi-Banach (sequence) lattice g ⊂ S satisfies theJackson’s inequality of order r if there exists C > such that σ ν ( t, s ) ≤ Ct − r k s k g for all s ∈ g. Definition 2.3.2.
Given r > , a quasi-Banach (sequence) lattice g ⊂ S satisfies theBernstein’s inequality of order r if there exists C > such that k t k g ≤ Ct r k t k f for all t ∈ Σ t,ν ∩ f. ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 5
We do not assume in the above definitions that g ֒ → f , but we need to assume t ∈ Σ t,ν ∩ f for Definition 2.3.2 to make sense.2.4. Weight functions for discrete Lorentz spaces.Definition 2.4.1.
We will denote by W the set of all continuous functions η :[0 , ∞ ) [0 , ∞ ) such thati) η (0) = 0 and lim t →∞ η ( t ) = ∞ ii) η is non-decreasingiii) η has the doubling property, that is, there exists C > such that η (2 t ) ≤ Cη ( t ) for all t > . A typical element of the class W is η ( t ) = t /p , 0 < p < ∞ . The functions in theclass W will be used to define general discrete Lorentz spaces. Occasionally, we willneed to assume a stronger condition on the function η ∈ W . For η ∈ W we define thedilation function M η ( s ) = sup t> η ( st ) η ( t ) , s > . Since η is non-decreasing, M η ( s ) ≤ < s ≤ Definition 2.4.2.
We say that η ∈ W + if η ∈ W and there exists s ∈ (0 , forwhich M η ( s ) < . Observe that for η ∈ W + and r > η r ∈ W + . Also, if η ∈ W and r > t r η ( t ) ∈ W + . Lemma 2.4.3.
Let η ∈ W + and take s as in the Definition 2.4.2. Then, there exists C > such that for all t > ∞ X j =0 η ( s j t ) ≤ Cη ( t ) . (2.4.1) Lemma 2.4.4.
Given η ∈ W + , there exists g ∈ C , g ∈ W + such that g ≈ η and g ′ ( t ) /g ( t ) ≈ /t , t > . General Discrete Lorentz Spaces.
We will define the discrete Lorentz spaceswe will work with. First, we recall some classical definitions (see e.g. [8] or [3]).For a sequence s = { s I } I ∈D ∈ S indexed by the countable set D , the non-increasingrearrangement of s with respect to a measure ν on D is s ∗ ν ( t ) = inf { λ > ν ( { I ∈ D : | s I | > λ } ) ≤ t } . For η ∈ W , ν a measure on D , and µ ∈ (0 , ∞ ], the discrete Lorentz space ℓ µη ( ν ) isthe set of all s = { s I } I ∈D ∈ S such that k s k ℓ µη ( ν ) := (cid:18)Z ∞ [ η ( t ) s ∗ ν ( t )] µ dtt (cid:19) /µ < ∞ , < µ < ∞ (2.5.1)and k s k ℓ ∞ η ( ν ) := sup t> η ( t ) s ∗ ν ( t ) < ∞ . EUGENIO HERN ´ANDEZ AND DANIEL VERA If η ( t ) = t /p , ≤ p < ∞ , then ℓ µη ( ν ) = ℓ p,µ ( ν ) are the classical (discrete) Lorentzspaces. For p = µ , ℓ p,p ( ν ) = ℓ p ( ν ) , < p < ∞ , are the spaces of sequences s ∈ S suchthat k s k ℓ p ( ν ) = X I ∈D | s I | p ν ( I ) ! /p . Notation . For ξ > η ∈ W , ˜ η ( t ) = t ξ η ( t ) ∈ W + and ℓ µ ˜ η ( ν ) will be denoted by ℓ µξ,η ( ν ). Proposition 2.5.1.
Let η ∈ W and ν a measure on D . For a sequence s = { s I } I ∈D ∈ S we have k s k ℓ ∞ η ( ν ) ≈ sup λ> λη ( ν ( { I ∈ D : | s I | > λ } )) . Moreover, if < µ < ∞ and η ∈ W + k s k ℓ µη ( ν ) ≈ (cid:18)Z ∞ [ λη ( ν ( { I ∈ D : | s I | > λ } ))] µ dλλ (cid:19) /µ . A sequence u = { u I } I ∈D ∈ S such that u I > I ∈ D will be called a weightsequence . Definition 2.5.2.
Let u = { u I } I ∈D be a weight sequence and ν a positive measure asdefined in Subsection 2.2. For < µ ≤ ∞ and η ∈ W define the space ℓ µη ( u , ν ) as theset of all sequences s = P I ∈D s I e I ∈ S such that k s k ℓ µη ( u ,ν ) := k{ u I s I } I ∈D k ℓ µη ( ν ) < ∞ . These spaces will be used in Subsections 2.6 and 2.7 to characterize Jackson andBernstein type inequalities in the setting of restricted non-linear approximation. Forapplications (see Subsections 2.8-2.11) we shall take u I = k e I k g , I ∈ D , where g is aquasi-Banach (sequence) lattice. Lemma 2.5.3.
Let u and ν as in Definition 2.5.2 and write Γ , u = P I ∈ Γ u − I e I , Γ ⊂ D and ν (Γ) < ∞ .(a) If η ∈ W , k Γ , u k ℓ ∞ η ( u ,ν ) = η ( ν (Γ)) .(b) If < µ < ∞ and η ∈ W , k Γ , u k ℓ µη ( u ,ν ) ≥ η ( ν (Γ)) , and if η ∈ W + , k Γ , u k ℓ µη ( u ,ν ) ≈ η ( ν (Γ)) . Jackson type inequalities.
We give equivalent conditions for some Jacksontype inequalities to hold in the setting of restricted non-linear approximation. Ourresult generalizes those obtained in [6] and [20] for restricted non-linear approximation,as well as those obtained in [19] and [15] for the case ν ( I ) = 1 for all I ∈ D (thecounting measure). Theorem 2.6.1.
Let ( f, ν ) be a standard scheme (see Definition 2.2.1)and let u = { u I } I ∈D be a weight sequence. Fix ξ > and µ ∈ (0 , ∞ ] . Then, for any function η ∈ W + the following are equivalent:1) There exists C > such that for all Γ ⊂ D with ν (Γ) < ∞ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X I ∈ Γ e I u I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ≤ Cη ( ν (Γ)) . ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 7 ℓ µξ,η ( u , ν ) ֒ → A ξµ ( f, ν ) .3)The space ℓ µξ,η ( u , ν ) satisfies Jackson’s inequality of order ξ , that is, there exists C > such that σ ν ( t, s ) f ≤ Ct − ξ k s k ℓ µξ,η ( u ,ν ) , for all s ∈ ℓ µξ,η ( u , ν ) . Taking η ( t ) = t /p , < p < ∞ , and u I = k e I k f in Theorem 2.6.1, condition 1)is called in [20] the (upper) p-Temlyakov property for f . In this case, ℓ µξ,η ( u , ν ) = ℓ q,µ ( u , ν ) with q = ξ + p .Taking ν as the counting measure on D we recover Theorem 3.6 in [15] from Theorem2.6.1.2.7. Bernstein type inequalities.
We give equivalent conditions for some Bernsteintype inequalities to hold in the setting of restricted non-linear approximation. Thisresult generalizes those obtained in [6] and [20] for restricted non-linear approximation,as well as those obtained in [19] and [15] for the case ν ( I ) = 1 for all I ∈ D (thecounting measure).We first begin with a representation theorem for the spaces A ξµ ( f, ν ). The prooffollows that in [27] replacing the counting measure by a general positive measure ν . Proposition 2.7.1.
Let ( f, ν ) be a standard scheme (see Definition 2.2.1). Fix ξ > and µ ∈ (0 , ∞ ] . The following statements are equivalenti) s ∈ A ξµ ( f, ν ) .ii) There exists s k ∈ Σ k ,ν ∩ f, k ∈ Z , such that s = P ∞ k = −∞ s k and { kξ k s k k f } k ∈ Z ∈ ℓ µ ( Z ) .Moreover, k s k A ξµ ( f,ν ) ≈ inf " ∞ X k = −∞ (2 kξ k s k k f ) µ /µ where the infimum is taken over all representations of s as in ii). Theorem 2.7.2.
Let ( f, ν ) be a standard scheme (see Definition 2.2.1)and let u = { u I } I ∈D be a weight sequence. Fix ξ > and µ ∈ (0 , ∞ ] . Then, for any function η ∈ W the following are equivalent:1) There exists C > such that for all Γ ⊂ D with ν (Γ) < ∞ , C η ( ν (Γ)) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X I ∈ Γ e I u I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f .
2) The space ℓ µξ,η ( u , ν ) satisfies Bernstein’s inequality of order ξ , that is, there exists C > such that k s k ℓ µξ,η ( u ,ν ) ≤ Ct ξ k s k f for all s ∈ Σ t,ν ∩ f. A ξµ ( f, ν ) ֒ → ℓ µξ,η ( u , ν ) . Taking η ( t ) = t /p , < p < ∞ , and u I = k e I k f in Theorem 2.7.2, condition 1)is called in [20] the (lower) p-Temlyakov property for f . In this case, ℓ µξ,η ( u , ν ) = ℓ q,µ ( u , ν ) with q = ξ + p . EUGENIO HERN ´ANDEZ AND DANIEL VERA
Theorem 2.7.2 generalizes Theorem 5 in [20] for a standard scheme. The proof ofthis theorem does not use the theory of real interpolation of quasi-Banach spaces; wewill, however, make use of it to shorten our proof.Taking ν as the counting measure on D we recover Theorem 4.2 in [15] from Theorem2.7.2.2.8. Restricted non-linear approximation and real interpolation.
It is wellknown that N -term approximation and real interpolation are interconnected. If theJackson and Bernstein’s inequalities hold for ν = counting measure, N -term approx-imation spaces are characterized in terms of interpolation spaces (see e.g. Theorem3.1 in [9] or Section 9, Chapter 7 in [8]).As pointed out in [6] the above mentioned theory can be developed in a more generalsetting. In particular, it can be done in the frame of the abstract scheme we haveintroduced in subsection 2.2. Below we state the results we need in this paper. Theproofs are straight-forward modifications of those given in the references cited in thefirst paragraph of this section. Theorem 2.8.1.
Let ( f, ν ) be a standard scheme. Suppose that the quasi-Banachlattice g ⊂ S satisfies the Jackson and Bernstein’s inequalities for some r > . Then,for < ξ < r and < µ ≤ ∞ we have A ξµ ( f, ν ) = ( f, g ) ξ/r,µ . It is not difficult to show that the spaces A rq ( f, ν ) , < r < ∞ , < q ≤ ∞ , satisfythe Jackson and Bernstein’s inequalities of order r , so that by Theorem 2.8.1, A ξµ ( f, ν ) = (cid:0) f, A rq ( f, ν ) (cid:1) ξ/r,µ for 0 < ξ < r and 0 < µ ≤ ∞ . From here, and using the reiteration theorem for realinterpolation we obtain the following result that will be used in the proof of Theorem2.7.2. Corollary 2.8.2.
Let < α , α < ∞ , < q, q , q ≤ ∞ and < θ < . Then, (cid:0) A α q ( f, ν ) , A α q ( f, ν ) (cid:1) θ,q = A αq ( f, ν ) , α = (1 − θ ) α + θα for a standard scheme ( f, ν ) . Sequence spaces associated with smoothness spaces.
A large number ofspaces used in Analysis are particular cases of the Triebel-Lizorkin and Besov spaces.Given s ∈ R , < p < ∞ , and 0 < r ≤ ∞ , the Triebel-Lizorkin spaces on R d aredenoted by F sp,r := F sp,r ( R d ) where s is a smoothness parameter, p measures integra-bility and r measures a refinement of smoothness. The reader can find the definitionof these spaces in [11, 12]. Lebesgue spaces L p ( R d ) = F p, , < p < ∞ , Hardy spaces H p ( R d ) = F p, , < p ≤
1, and Sobolev spaces W sp ( R d ) = F sp, , s > , < p < ∞ , areincluded in this collection.Given s ∈ R , < p, r ≤ ∞ , the Besov spaces on R d are denoted by B sp,r := B sp,r ( R d )with an interpretation of the parameters as in the case of the Triebel-Lizorkin spaces.These spaces include the Lipschitz classes (see [12]). ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 9
There are characterizations of F sp,r and B sp,r in terms of sequence spaces. Suchcharacterizations were given first in [11] using the ϕ -transform. Wavelet bases withappropriate regularity and moment conditions also provide such characterizations.A brief description of wavelet bases in R d follows. Let D be the set of dyadic cubesin R d given by Q j,k = 2 − j ([0 , d + k ) , j ∈ Z , k ∈ Z d . A finite collection of functions Ψ = { ψ (1) , . . . , ψ ( L ) } ⊂ L ( R d ) with L = 2 d − W := { ψ ( ℓ ) Q j,k ( x ) := 2 jd ψ ( ℓ ) (2 j x − k ) : Q j,k ∈ D , ℓ = 1 , , . . . , L } is an orthonormal basis for L ( R d ). This is the definition that appears in [23]. Thereader can consult properties of wavelets in [26], [7], [16] and [24]. Definition 2.9.1.
Given s ∈ R , < p < ∞ and < r ≤ ∞ , we let f sp,r be the spaceof sequences s = { s Q } Q ∈D such that k s k f sp,r := (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)"X Q ∈D ( | Q | − s/d +1 /r − / | s Q | χ ( r ) Q ( · )) r /r (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L p ( R d ) < ∞ where χ ( r ) Q ( · ) = χ Q ( · ) | Q | − /r and χ Q ( · ) denotes the characteristic function of Q . Definition 2.9.2.
Given s ∈ R , < p, r ≤ ∞ , we let b sp,r be the space of sequences s = { s Q } Q ∈D such that k s k b sp,r := X j ∈ Z X | Q | =2 − jd ( | Q | − s/d +1 /p − / | s Q | ) p r/p /r < ∞ with the obvious modifications when p, r = ∞ . With appropriate conditions in the elements of a wavelet family Ψ = { ψ (1) , . . . , ψ ( L ) } ,( L = 2 d −
1) it can be shown that W is an unconditional basis of F sp,r or B sp,r and if f = P Lℓ =1 P Q ∈D s ℓQ ψ ( ℓ ) Q , then k f k F sp,r ≈ L X ℓ =1 (cid:13)(cid:13) { s ℓQ } Q ∈D (cid:13)(cid:13) f sp,r and k f k B sp,r ≈ L X ℓ =1 (cid:13)(cid:13) { s ℓQ } Q ∈D (cid:13)(cid:13) b sp,r . (2.9.1)Conditions on Ψ for these equivalences to hold can be found in [26], [16], [1], [22], [23].When a wavelet family Ψ provides an unconditional basis for F sp,r or B sp,r , with equiv-alences as in (2.9.1), we shall say that Ψ is admissible for F sp,r or B sp,r , respectively.The equivalences (2.9.1) allow us to work at the sequence level. We shall drop thesum over ℓ since it only changes the constants in the computations below. The resultsproved for sequence spaces f sp,r or b sp,r can be transferred to F sp,q or B sp,r by the abstracttransference framework developed in [13].We notice that the Triebel-Lizorkin and Besov spaces characterized as in (2.9.1) arecalled homogeneous, been often denoted by ˙ F sp,r and ˙ B sp,r . The non-homogeneous caserequires small modifications. Also minor modifications will allow for the anisotropicspaces as considered in [13], or the spaces defined by wavelets on bounded domains.We restrict ourselves to the cases characterized by (2.9.1). Restricted approximation for Triebel-Lizorkin sequence spaces.
As con-sequence of the theorems developed in Sections 2.6 and 2.7 we will obtain results forrestricted approximation in Triebel-Lizorkin sequence spaces. When coupled with theabstract transference framework developed in [13], our results generalizes those in [6]and, with minor modifications, those obtained in [18].
Lemma 2.10.1.
Let Γ ⊂ D (not necessarily finite), x ∈ ∪ Q ∈ Γ Q , and γ = 0 . Define S γ Γ ( x ) = X Q ∈ Γ | Q | γ χ Q ( x ) . i) If γ > and there exists Q x , the biggest cube in Γ that contains x , then S γ Γ ( x ) ≈| Q x | γ χ Q x ( x ) = | Q x | γ ii) If γ < and there exists Q x , the smallest cube in Γ that contains x , then S γ Γ ( x ) ≈ | Q x | γ χ Q x ( x ) = | Q x | γ . The smallest cube Q x from Γ that contains x ∈ ∪ Q ∈ Γ Q has been used by otherauthors in the context of non-linear approximation with wavelet basis (see [17], [6],[13], [14]). As far as we know, the biggest cube Q x from Γ that contains x ∈ ∪ Q ∈ Γ Q has not been used before. Theorem 2.10.2.
Let s , s ∈ R , < p , p < ∞ , < q , q ≤ ∞ . For α ∈ R and Γ ⊂ D define ν α (Γ) = P Q ∈ Γ | Q | α . Suppose ν α (Γ) < ∞ . Then, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X Q ∈ Γ e Q k e Q k f s p ,q (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f s p ,q ≈ [ ν α (Γ)] /p (2.10.1) if and only if α = 1 and α = p ( s − s d − p ) + 1 or α = 1 , s − s d = p and p = q . Theorems 2.6.1 and 2.7.2 with η ( t ) = t /p and u Q = k e Q k f s p ,q together withTheorem 2.10.2 show that non-linear approximation with error measured in f s p ,q whenthe basis is normalized in f s p ,q is related to the use of the measure ν α ( Q ) = | Q | α , Q ∈ D , α = p ( s − s d − p ), to control the number of terms in the approximation.Notice that no role is played by the second smoothness parameters q , q .Theorems 2.6.1 and 2.7.2 together with Theorem 2.10.2 also allow us to identifythe restricted approximation spaces in the Triebel-Lizorkin setting as discrete Lorentzspaces. Corollary 2.10.3.
Let s , s ∈ R , < p , p < ∞ , < q , q ≤ ∞ and define α = p ( s − s d − p ) + 1 . For Γ ⊂ D define ν α (Γ) = P Q ∈ Γ | Q | α . Let ξ > and µ ∈ (0 , ∞ ] . If α = 1 , A ξµ ( f s p ,q , ν α ) = ℓ τ,µ ( u , ν α ) , where τ = ξ + p and u = {k e Q k f s p ,q } Q ∈D . If α = 1 the result holds with p = q . For particular values of the parameters, the discrete Lorentz spaces that appear inCorollary 2.10.3 can be identified as Besov spaces.
Lemma 2.10.4.
Let s , s ∈ R , < p , p < ∞ , < q ≤ ∞ and define α = p ( s − s d − p ) + 1 . For Γ ⊂ D define ν α (Γ) = P Q ∈ Γ | Q | α . Given τ ∈ (0 , ∞ ) we have ℓ τ,τ ( u , ν α ) = b γτ,τ , (with equal quasi-norms) where u = {k e Q k f s p ,q } Q ∈D and γ = s + d ( τ − p )(1 − α ) . ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 11
Remark 2.10.5.
If we consider the point of view of [20] , then we can only prove ℓ τ,τ ( u , ν α ) = b γτ,τ ∩ f s p ,q and the equivalence of quasi-norms holds if we take k·k b γτ,τ inthe right-hand side.When τ < p and γd − τ = s d − p it is known that b γτ,τ ֒ → f s p ,q (see [10] or [4] ).This situation occurs when α = 0 (the counting measure) but it is not true in ourmore general situation. The following result identifies certain non-linear approximation spaces in the re-stricted setting, when the error is measured in Triebel-Lizorkin spaces, as Besov spaces.It is obtained as an easy corollary to Lemma 2.10.4 and Corollary 2.10.3.
Corollary 2.10.6.
Let s , s ∈ R , < p , p < ∞ , < q ≤ ∞ and define α = p ( s − s d − p ) + 1 . For Γ ⊂ D define ν α (Γ) = P Q ∈ Γ | Q | α . Given ξ > define τ by τ = ξ + p . If α = 1 , A ξτ ( f s p ,q , ν α ) = b γτ,τ (equivalent quasi-norms) , where γ = s + dξ (1 − α ) . If α = 1 the result holds with γ = s and p = q . Remark 2.10.7.
If we were to apply Theorem 1 in [20] we will obtain A ξτ ( f s p ,q , u , ν α ) = b γτ,τ ∩ f s p ,q with equivalence of quasi-norms, as in Remark 2.10.5. The results obtained in [6] for restricted non-linear approximation with waveletsin the Hardy space H p , < p < ∞ , when the wavelets coefficients are restrictedto H r , < r < ∞ , are simple consequences of the above results and the abstracttransference framework developed in [13]. To see this, notice that the sequence spacesassociated to H p and H r ( r, p as above) are f p, and f r, , respectively.Thus, for a wavelet basis W = { ψ ( ℓ ) Q : Q ∈ D , ℓ = 1 , . . . , L } , ( L = 2 d −
1) admissiblefor H p and B γτ,τ , A ξτ ( H p , W , ν α ) = B γτ,τ , (2.10.2)where γ = dpr ξ , τ defined by τ = ξ + p , and α = 1 − p/r ( = 1). This is Corollary 6.3in [6]. Notice that A ξτ ( H p , W , ν α ) corresponds to an approximation space where thewavelet coefficients are normalized in H r . In the above notation we have emphasizethat the approximation spaces are defined using wavelet basis.In this situation, The Jackson and Bernstein’s inequalities (Theorems 5.1 and 5.2in [6]) follow from (2.10.2) and the fact that the approximation spaces always satisfythe Jackson and Bernstein’s inequalities.The other situation considered in [6] is B p := B p,p , < p < ∞ , when the wavelet co-efficients are restricted in H r , < r < ∞ . In this case, the sequence spaces associatedto B p,p = F p,p and H r are f p,p and f r, , respectively. Corollary 2.10.6 then produces A ξτ ( B p , W , ν α ) = B γτ,τ , (2.10.3)where γ = dpr ξ , τ defined by τ = ξ + p , with ν α and W as before. This is moregeneral than Corollary 6.1 in [6] and a comparison with (2.10.2) proves immediatelya more general version of Theorem 6.3 in [6]. Of course, the Jackson and Bernstein’sinequalities of Theorems 5.4 and 5.5 in [6] also follow from our results.To show an example not treated in [6] consider the wavelet orthonormal basis W = { ψ ( ℓ ) Q : Q ∈ D , ℓ = 1 , . . . , L } ( L = 2 d −
1) admissible for the Sobolev space W s , s > We want to measure the error in W s but we restrict the wavelet coefficients to L .Since the sequence spaces associated to W s and L are f s , and f , , Corollary 2.10.6and the abstract framework of [13] gives A ξτ ( W s , W , ν α ) = b γτ,τ where γ = s + dξ (1 − α ), α = − s/d and τ defined by τ = ξ + .We remark that defining appropriate sequence spaces, a little more work will showthe results proved in [18] for the anisotropic case.2.11. Application to Real Interpolation.
Once the restricted approximation spacesfor Triebel-Lizorkin sequence spaces have been identified (see Corollary 2.10.6) we canuse Theorem 2.8.1 to obtain results about real interpolation. This method has beenused before (see [9] or [13]). But in the classical case, the parameters of the spacesinterpolated are restricted. With the theory of restricted approximation we will proveinterpolation results for a much larger set of parameters.
Theorem 2.11.1.
Let s ∈ R , < p < ∞ , < q ≤ ∞ . For < τ < p , < θ < and γ = s ( γ ∈ R ) we have (cid:0) f sp,q , b γτ,τ (cid:1) θ,τ θ = b (1 − θ ) s + θγτ θ ,τ θ with τ θ = 1 − θp + θτ . Remark 2.11.2.
Although the Theorem is presented as a result about interpolationof Triebel-Lizorkin and Besov (sequence) spaces, it is a result about interpolation ofTriebel-Lizorkin sequence spaces, since b γτ,τ = f γτ,τ . Thus, the result can be stated as (cid:0) f sp,q , f γτ,τ (cid:1) θ,τ θ = f (1 − θ ) s + θγτ θ ,τ θ with τ θ = 1 − θp + θτ . (2.11.1) Remark 2.11.3.
Notice that we do not need the restriction γd − τ = sd − p characteristicof this type of results when classical non-linear approximation is used (see e.g. [9] or [13] ). Remark 2.11.4.
By the transference framework designed in [13] , the result of The-orem 2.11.1 can be translated to a result for (homogeneous) Triebel-Lizorkin spaces.The non-homogeneous case and the case of bounded domains can also be obtained withminor modifications in the proof.
Remark 2.11.5.
The merit of Theorem 2.11.1 is that proves results for a large setof parameters using the theory of approximation. Nevertheless, many (but not all, asfar as we know) have already been proved. One can read from Theorem 3.5 in [4] thefollowing result: (cid:0) F s p ,q , F s p ,q (cid:1) θ,p = F sp,p = B sp,p (2.11.2) when p i < q i , i = 0 , , p = − θp + θp , s = (1 − θ ) s + θs and s = s . Comparingwith (2.11.1) we see that (2.11.2) has a larger set of parameters in the second space,while (2.11.1) does not have the restriction p < q that is required in (2.11.2). Both ofthese shortcomings are due to the methods of the proofs. On the other hand, Theorem2.42/1 (page 184) of [28] shows (2.11.2) without p i < q i but assuming < p i , q i < ∞ . ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 13 Proofs
Weight functions for discrete Lorentz spaces (proofs).Proof of Lemma 2.4.3 . Let δ := M η ( s ) <
1. By definition of M η we have1 > δ ≥ η ( s j +10 t ) η ( s j t ) for all j = 0 , , , . . . Therefore, ∞ X j =0 η ( s j t ) ≤ ∞ X j =0 δ j η ( t ) = η ( t ) 11 − δ . (cid:4) Proof of Lemma 2.4.4 . Define g ( t ) = R t η ( s ) s ds . With s as in Definition 2.4.2 g ( t ) = ∞ X j =0 Z s j ts j +10 t η ( s ) s ds ≤ ∞ X j =0 η ( s j t ) log( s − ) ≤ Cη ( t ) log( s − )by Lemma 2.4.3 ( C = − δ , see the proof of Lemma 2.4.3). On the other hand g ( t ) ≥ Z tt/ η ( s ) s ds ≥ η ( t/
2) log 2 ≥ Dη ( t ) log 2by the doubling property of η . This shows C η ( t ) ≤ g ( t ) ≤ C η ( t ) , t ∈ (0 , ∞ ) (3.4.1)with 0 < C ≤ C < ∞ . As an alternative proof one can see that η satisfies thehypotheses of Lemma 1.4 in [21] (p. 54) to conclude g ≈ η . The function g isclearly non-decreasing and (3.4.1) shows that g ∈ W . It is clear that g ∈ C with g ′ ( t ) = η ( t ) /t . Thus g ′ ( t ) g ( t ) = η ( t ) /tη ( t ) = 1 t . It remains to prove that g ∈ W + . To prove this, observe that a function η ∈ W is anelement of W + if and only if i η := lim t → + log M η ( t )log t > i η is called the lower dilation (Boyd) index of η - see [3]). Using (3.4.1) i g = lim t → + log M g ( t )log t ≥ lim t → + log( C C M η ( t ))log t = lim t → + log( M η ( t ))log t = i η and, similarly i g ≤ lim t → + log( C C M η ( t ))log t = i η . Thus, i g = i η > g ∈ W + . (cid:4) General discrete Lorentz spaces (proofs).Proof of Proposition 2.5.1.
The case µ = ∞ follows from part iii) of Proposition2.2.5 in [5]. For 0 < µ < ∞ , let w ( t ) = [ η ( t )] µ /t , 0 < t < ∞ . Writing λ ν ( t, s ) = ν ( { I ∈ D : | s I | > t } ) for the distribution function of s with respect to the measure ν and W( s ) = R s w ( t ) dt , 0 < s < ∞ , part ii) of Proposition 2.2.5 in [5] gives k s k ℓ µη ( ν ) = (cid:18)Z ∞ µt µ W( λ ν ( t, s )) dtt (cid:19) /µ . Since η ∈ W + , η µ satisfies the hypothesis of Lemma 1.4 in [21] (p. 54) so that weconclude W( s ) = Z s η ( t ) µ t dt ≈ [ η ( s )] µ (see also the proof of Lemma 2.4.4 and the comment that follows Definition 2.4.2).This proves the result. (cid:4) Proof of Lemma 2.5.3. (a) Writing 1 Γ , u = P I ∈D s I e I we have s I = u − I for all I ∈ Γ and s I = 0 if I Γ. Thus, u I s I = 1 for all I ∈ Γ and u I s I = 0 if I Γ. Thisimplies { u I s I } ∗ ν ( t ) = (cid:26) , < t < ν (Γ)0 , t ≥ ν (Γ) (cid:27) . (3.5.1)By Definition 2.5.2 k Γ , u k ℓ ∞ η ( u ,ν ) = sup 2) log 2 ≥ Cη ( ν (Γ))since η is doubling. For the reverse inequality, since η ∈ W + , by Proposition 2.5.1 weobtain k Γ , u k ℓ µη ( u ,ν ) ≈ (cid:18)Z [ λη ( ν (Γ))] µ dλλ (cid:19) /µ ≈ η ( ν (Γ)) . (cid:4) Jackson type inequalities (proofs). ⇒ 3) This is immediate since A ξµ ( ν ) ֒ →A ξ ∞ ( ν ) and 3) is equivalent to ℓ µξ,η ( u , ν ) ֒ → A ξ ∞ ( f, ν ).3) ⇒ 1) Let 0 < s < M η ( s ) < η ∈ W + .Let Γ ⊂ D with ν (Γ) < ∞ and write 1 Γ := 1 Γ , u = P I ∈ Γ e I u I . By Lemma 1 in [20] (seealso the proof of Theorem 2.1 in [13]), for Λ = Γ ⊂ D one can find a subset Λ ⊂ Λ with ν (Λ ) ≤ s ν (Λ ) such that k Λ − Λ k f ≈ σ ν ( s ν (Λ ) , Λ ) . We repeat this argument to find nested subsetsΓ = Λ ⊃ Λ ⊃ . . . ⊃ Λ j ⊃ Λ j +1 ⊃ . . . ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 15 such that ν (Λ j +1 ) ≤ s ν (Λ j ) and (cid:13)(cid:13) Λ j − Λ j +1 (cid:13)(cid:13) f ≈ σ ν ( s ν (Λ j ) , Λ j ) , j = 0 , , , . . . By the ρ -power triangle inequality for f we get k Γ k ρf ≤ ∞ X j =0 (cid:13)(cid:13) Λ j − Λ j +1 (cid:13)(cid:13) ρf ≈ ∞ X j =0 σ ρν ( s ν (Λ j ) , Λ j ) . Using the hypothesis and Lemma 2.5.3 we obtain σ ν ( s ν (Λ j ) , Λ j ) ≤ C [ s ν (Λ j )] − ξ (cid:13)(cid:13) Λ j (cid:13)(cid:13) ℓ µξ,η ( ν ) ≈ η ( ν (Λ j )) . Concatenating these inequalities we deduce k Γ k f . " ∞ X j =0 η ρ ( ν (Λ j )) /ρ . " ∞ X j =0 η ρ ( s j ν (Λ )) /ρ . η ( ν (Λ )) = η ( ν (Γ))by Lemma 2.4.3, since η and η ρ belong to W + .1) ⇒ 2) By Lemma 2.4.4 we may assume η ∈ C and η ′ ( t ) /η ( t ) ≈ /t , t > 0. Westart by bounding σ ν ( t, s ) for s ∈ ℓ µξ,η ( u , ν ). Recall that d := { s I u I } I ∈D ∈ ℓ µξ,η ( ν ).Since ν ( { I ∈ D : | u I s I | > d ∗ ν ( t ) } ) ≤ t we have σ ν ( t, s ) = inf t ∈ Σ t,ν k s − t k f ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X | u I s I |≤ d ∗ ν ( t ) s I e I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f . For j = 0 , , , . . . let Λ j = { I ∈ D : 2 − j − d ∗ ν ( t ) < | s I u I | ≤ − j d ∗ ν ( t ) } . The ρ -powertriangle inequality and the monotonicity property of f , together with the hypothesis,imply [ σ ν ( t, s )] ρ ≤ ∞ X j =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X I ∈ Λ j s I e I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρf = ∞ X j =0 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X I ∈ Λ j s I u I e I u I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρf ≤ C ∞ X j =0 [2 − j d ∗ ν ( t )] ρ η ρ ( ν (Λ j )) ≤ C Z d ∗ ν ( t )0 λ ρ [ η ( ν ( { I ∈ D : | s I u I | > λ } ))] ρ dλλ . Applying part 2 of Lemma 2 in [20] with F ( λ ) = λ ρ ρ and G ( λ ) = [ η ( λ )] ρ yields[ σ ν ( t, s )] ρ ≤ ρ [ d ∗ ν ( t ) η ( t )] ρ + 1 ρ Z ∞ t [ d ∗ ν ( s )] ρ dη ρ ( s ) ≈ [ d ∗ ν ( t ) η ( t )] ρ + Z ∞ t [ d ∗ ν ( s )] ρ η ρ ( s ) dss where we have used η ′ ( s ) /η ( s ) ≈ /s . Therefore σ ν ( t, s ) . d ∗ ν ( t ) η ( t ) + (cid:18)Z ∞ t [ d ∗ ν ( s ) η ( s )] ρ dss (cid:19) /ρ . Thus, k s k µ A ξµ ( ν ) = Z ∞ [ t ξ σ ν ( t, s )] µ dtt . Z ∞ [ t ξ η ( t ) d ∗ ν ( t )] µ dtt + Z ∞ " t ξ (cid:18)Z ∞ t [ d ∗ ν ( s ) η ( s )] ρ dss (cid:19) /ρ µ dtt := I + II. The first term, I , is precisely k s k µℓ µξ,η ( u ,ν ) . For II use Hardy’s inequality (see [3], p.124)with a ρ such that µ/ρ > f satisfiesthe ρ -power triangle inequality it satisfies the ρ ′ -power triangle inequality for any0 < ρ ′ ≤ ρ ) to obtain II /µ = "Z ∞ t ξµ (cid:18)Z ∞ t [ d ∗ ν ( s ) η ( s )] ρ dss (cid:19) µ/ρ dtt /µ ≤ (cid:20) ρξ Z ∞ [ d ∗ ν ( s ) η ( s )] µ s ξµ dss (cid:21) /µ = C k s k ℓ µξ,η ( u ,ν ) , This proves the result. (cid:4) Bernstein type inequalities (proofs).Proof of Proposition 2.7.1. i) ⇒ ii) Let s ∈ A ξµ ( f, ν ). Choose ϕ k ∈ Σ k − ,ν such that k s − ϕ k k f ≤ σ ν (2 k − , s ). Let s k = ϕ k − ϕ k − , so that s k ∈ Σ k ,ν . Since s ∈ A ξµ ( f, ν ) we have σ ν (2 k − , s ) → k → ∞ ; the assumption f ֒ → S implieslim k →∞ ϕ k = s in D (term by term).On the other hand lim k →−∞ ϕ k = 0 since ν (supp ϕ k ) → k → −∞ . Thus, s = lim k →∞ ϕ k = ∞ X k = −∞ s k . Now, k s k k ρf ≤ k s − ϕ k k ρf + k s − ϕ k − k ρf ≤ · ρ [ σ ν (2 k − , s )] ρ . Therefore, X k ∈ Z [2 kξ k s k k f ] µ ≤ C X k ∈ Z [2 kξ σ ν (2 k − , s )] µ ≈ k s k µ A ξµ ( ν ) , by the discrete characterization of the restricted approximation spaces given in Sub-section 2.2. It is easy to see that the result also holds for µ = ∞ .ii) ⇒ i) Observe that P ℓ − k = −∞ s k ∈ P ℓ ,ν since each s k ∈ Σ k ,ν . Take ρ such that0 < ρ < µ and k·k f satisfies the ρ -power triangle inequality. We have[ σ ν (2 ℓ , s )] ρ ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s − ℓ − X k = −∞ s k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ρf ≤ ∞ X k = ℓ k s k k ρf . ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 17 With p = µ/ρ > 1, (here 0 < µ < ∞ ) and u > u < ξρ we have k s k µ A ξµ ( ν ) ≈ ∞ X ℓ = −∞ [2 ℓξ σ ν (2 ℓ , s )] µ ≤ ∞ X ℓ = −∞ [2 ℓξ ( ∞ X k = ℓ k s k k ρf ) /ρ ] µ = ∞ X ℓ = −∞ ℓξµ ( ∞ X k = ℓ − ku ku k s k k ρf ) p ≤ ∞ X ℓ = −∞ ℓξµ ( ∞ X k = ℓ − kup ′ ) p/p ′ ( ∞ X k = l kup k s k k µf ) ≈ X ℓ ∈ Z ℓ ( ξµ − up ) ∞ X k = ℓ kup k s k k µf ! = X k ∈ Z kup k s k k µf ( k X ℓ = −∞ ℓ ( ξµ − up ) ) ≈ X k ∈ Z kξµ k s k k µf . Since the last expression is finite by hypothesis, we have proved s ∈ A ξµ ( f, ν ) for0 < µ < ∞ . For µ = ∞ we have k s k A ξ ∞ ( ν ) ≈ sup ℓ ∈ Z ℓξ σ ν (2 ℓ , s ) ≤ sup ℓ ∈ Z ℓξ ( ∞ X k = ℓ k s k k ρf ) /ρ = sup ℓ ∈ Z ℓξ ( ∞ X k = ℓ − kξρ kξρ k s k k ρf ) /ρ ≤ (sup k ∈ Z kξ k s k k f ) sup ℓ ∈ Z ℓξ ( ∞ X k = ℓ − kξρ ) /ρ ≈ sup k ∈ Z kξ k s k k f . (cid:4) Proof of Theorem 2.7.2. ⇒ 2) Let s ∈ Σ t,ν ∩ f and write s = { s I } I ∈D . Given0 < τ ≤ ν (Γ) choose Λ τ = { I ∈ Γ : | u I s I | ≥ d ∗ ν ( τ ) } where d = { u I s I } I ∈D . We have τ ≤ ν (Λ τ ) (see (4) in [20] or [2]). Applying the hypothesis 1) and the monotonicityof f we obtain d ∗ ν ( τ ) η ( τ ) ≤ d ∗ ν ( τ ) η ( ν (Λ τ )) . d ∗ ν ( τ ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X I ∈ Λ τ e I u I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X I ∈ Λ τ s I e I (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f . k s k f . Since ν ( { I ∈ D : | s I u I | > } ) = ν (Γ) ≤ t we have d ∗ ν ( τ ) = 0 for τ ≥ t . Thus, k s k µℓ µξ,η ( u ,ν ) = Z t [ τ ξ η ( τ ) d ∗ ν ( τ )] µ dττ . k s k µf Z t τ ξµ dττ ≈ t ξµ k s k µf . The case µ = ∞ is treated similarly.2) ⇒ 1) Let 1 Γ := 1 Γ , u = P I ∈ Γ e I u I and ν (Γ) = t , so that 1 Γ ∈ Σ t,ν . We may assume1 Γ ∈ f , since otherwise the right hand side of 1) is ∞ , and the result is trivially true.Hypothesis 2) gives k Γ k f & t − ξ k Γ k ℓ µξ,η ( u ,ν ) & η ( ν (Γ)) , where the last inequality is due to Lemma 2.5.3. ⇒ 2) For s ∈ Σ t,ν ∩ f , σ ν ( τ, s ) = 0 if τ ≥ t . Thus, by 3) k s k ℓ µξ,η ( u ,ν ) . k s k A ξµ ( ν ) = (cid:18)Z t [ τ ξ σ ν ( τ, s )] µ dττ (cid:19) /µ . k s k f (cid:18)Z t τ ξµ dττ (cid:19) /µ = t ξ k s k f , where we have used σ ν ( τ, s ) ≤ k s k f for all τ > ⇒ 3) We have already proved that 1) ⇔ ξ, µ ,then 2) holds for all ˜ ξ > µ ∈ (0 , ∞ ]. For any ˜ ξ > ρ such that ℓ µ ˜ ξ,η ( u , ν ) satisfies the ˜ ρ -power triangular inequality. By Proposition 2.7.1 we can find s k ∈ Σ k ,ν ∩ f , k ∈ Z , such that s = P k ∈ Z s k (in D ) and k s k A ˜ ξ ˜ ρ ( ν ) ≈ X k ∈ Z [2 k ˜ ξ k s k f ] ˜ ρ ! / ˜ ρ . Applying hypothesis 2) to ℓ µ ˜ ξ,η ( u , ν ) we obtain k s k ˜ ρℓ µ ˜ ξ,η ( u ,ν ) . X k ∈ Z [ k s k k ℓ µ ˜ ξ,η ( u ,ν ) ] ˜ ρ . X k ∈ Z (2 k ˜ ξ k s k k f ) ˜ ρ ≈ k s k ˜ ρ A ˜ ξ ˜ ρ ( ν ) . This means that for µ ∈ (0 , ∞ ] and any ˜ ξ > A ˜ ξ ˜ ρ ( ν ) ֒ → ℓ µ ˜ ξ,η ( u , ν ) , (3.7.1)where ˜ ρ is the exponent of the ˜ ρ -power triangle inequality for ℓ µ ˜ ξ,η ( u , ν ). From Corol-lary 2.8.2, for ξ = ( ξ + ξ ) / ρ ∈ (0 , 1] we have( A ξ ρ ( ν ) , A ξ ρ ( ν )) / ,µ = A ξµ ( ν ) . Applying (3.7.1) with ˜ ξ = ξ , first, then ˜ ξ = ξ and ρ = min { ˜ ρ , ˜ ρ } we obtain A ξµ ( ν ) = ( A ξ ρ ( ν ) , A ξ ρ ( ν )) / ,µ ֒ → ( ℓ µξ ,η ( u , ν ) , ℓ µξ ,η ( u , ν )) / ,µ = ℓ µξ,η ( u , ν )where the last equality is a result in real interpolation of discrete Lorentz spaces thatcan be found in [25] (Theorem 3). (cid:4) Restricted approximation for Triebel-Lizorkin sequence spaces (proofs).Proof of Lemma 2.10.1. i) It is clear that | Q x | γ χ Q x ( x ) ≤ S γ Γ ( x ) since the righthand side of this inequality contain at least the cube Q x (and possibly more). Forthe reverse inequality we enlarge the sum defining S γ Γ ( x ) to include all dyadic cubescontained in Q x . Therefore, S γ Γ ( x ) ≤ X Q ⊂ Q x : Q ∈D | Q | γ = ∞ X j =0 (2 − jd | Q x | ) γ = | Q x | γ ∞ X j =0 − jdγ ≈ | Q x | γ since γ > ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 19 ii) It is clear that | Q x | γ χ Q x ( x ) ≤ S γ Γ ( x ) since the right hand side of this inequalitycontains at least the cube Q x (and possibly more). For the reverse inequality, weenlarge the sum defining S γ Γ ( x ) to include all dyadic cubes containing Q x . Therefore, S γ Γ ( x ) ≤ X Q ⊃ Q x : Q ∈D | Q | γ = ∞ X j =0 (2 jd | Q x | ) γ = | Q x | γ ∞ X j =0 jdγ ≈ | Q x | γ since γ < (cid:4) Proof of Theorem 2.10.2. We start by proving (2.10.1). Write f := f s p ,q and f := f s p ,q to simplify notation in this proof. By Definition 2.9.1 we have k e Q k f = | Q | − s /d +1 /p − / and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X Q ∈ Γ e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = Z R d "X Q ∈ Γ | Q | s − s d q | Q | − q /p χ Q ( x ) p /q dx /p = Z R d "X Q ∈ Γ ( | Q | α − p χ Q ( x )) q p /q dx /p . (3.10.1)Consider first the case α > 1. In this case, since ν α (Γ) < ∞ , the biggest Q x containedin Γ exists for all x ∈ ∪ Q ∈ Γ Q . Applying Lemma 2.10.1, part i), first with γ = α − p q > γ = α − > 0, we obtain "X Q ∈ Γ | Q | α − p q χ Q ( x ) p /q ≈ | Q x | α − χ Q x ( x ) ≈ X Q ∈ Γ | Q | α − χ Q ( x )for all x ∈ ∪ Q ∈ Γ Q . From (3.10.1) we deduce (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X Q ∈ Γ e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ≈ Z R d X Q ∈ Γ | Q | α − χ Q ( x ) dx ! /p = X Q ∈ Γ | Q | α ! /p = [ ν α (Γ)] /p . Consider now the case α < 1. If α ≤ 0, since ν α (Γ) < ∞ , the smallest cube Q x contained in Γ exists for all x ∈ ∪ Q ∈ Γ Q (notice that α = 0 is the classical case ofcounting measure). If 0 < α < E α of all x ∈ ∪ Q ∈ Γ Q forwhich Q x does not exists has measure zero. To see this, write D k = { Q ∈ D : | Q | =2 − kd , k ∈ Z } . Then, for all m ≥ E α ⊂ ∪ k ≥ m ∪ Q ∈ Γ ∩D k Q ; therefore | E α | ≤ X k ≥ m X Q ∈ Γ ∩D k | Q | = X k ≥ m X Q ∈ Γ ∩D k | Q | α | Q | − α ≤ ν α (Γ) X k ≥ m − kd (1 − α ) ≈ ν α (Γ)2 − md (1 − α ) since 1 − α > 0. Letting m → ∞ we deduce | E α | = 0. Apply Lemma 2.10.1, part ii), first with γ = α − p q < γ = α − < "X Q ∈ Γ | Q | α − p q χ Q ( x ) p /q ≈ | Q x | α − χ Q x ( x ) ≈ X Q ∈ Γ | Q | α − χ Q ( x )for all x ∈ ∪ Q ∈ Γ Q if α ≤ x ∈ ∪ Q ∈ Γ Q \ E α if 0 < α < 1. In any case, from(3.10.1) we deduce (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X Q ∈ Γ e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f ≈ Z R d X Q ∈ Γ | Q | α − χ Q ( x ) dx ! /p = X Q ∈ Γ | Q | α ! /p = [ ν α (Γ)] /p . For α = 1 the set E of all x ∈ ∪ Q ∈ Γ Q for which Q x exists has also measure zero.Indeed | E | ≤ X k ≥ m X Q ∈ Γ ∩D k | Q | = X k ≥ m ν (Γ ∩ D k )and the last sum tends to zero as m → ∞ since they are the tails of the convergentsum P k ≥ ν (Γ ∩ D k ) ≤ ν (Γ) < ∞ . Suppose now that (2.10.1) holds. For N ∈ N and L = 2 l consider the set Γ N,L = { [0 , L ] d + Lj : j ∈ N d , ≤ | j | < N } of N d disjoint dyadic cubes of size length L . Forthis collection we have ν α (Γ N,L ) = X Γ N,L | Q | α = ( L α N ) d . (3.10.2)Also (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ N,L e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = (cid:18)Z R d h S γ Γ N,L ( x ) i p /q dx (cid:19) /p with γ = ( s − s d − p ) q . Since S γ Γ N,L ( x ) = L dγ P Γ N,L χ Q ( x ) = L dγ χ [0 ,NL ] d ( x ) we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ N,L e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = L dγ/q ( LN ) d/p = L d (cid:16) γq + 1 p (cid:17) N d/p . (3.10.3)Choose N, N ′ ∈ N , L = 2 l , L ′ = 2 l ′ such that L α N = ( L ′ ) α N ′ so that (3.10.2) implies ν α (Γ N,L ) = ν α (Γ N ′ ,L ′ ) . By (2.10.1) and (3.10.3) we deduce L d (cid:16) γq + 1 p (cid:17) N d/p ≈ ( L ′ ) d (cid:16) γq + 1 p (cid:17) ( N ′ ) d/p ⇔ (cid:18) LL ′ (cid:19) d (cid:16) γq + 1 − αp (cid:17) ≈ . This forces γq = α − p , or equivalently s − s d − p = α − p as desired.For α = 1 we still have to prove that p = q . Let N ∈ N and Γ N = { Q ⊂ [0 , d :2 − Nd < | Q | ≤ } . We have ν (Γ N ) = N and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ Γ N e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = Z R d X Γ N χ Q ( x ) ! p /q dx /p = N /q . (3.10.4) ESTRICTED NON-LINEAR APPROXIMATION IN SEQUENCE SPACES 21 For the same N ∈ N take ˜Γ N = { [0 , d + j −→ e : 0 ≤ j < N } so that ν (˜Γ N ) = N and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X Q ∈ ˜Γ N e Q k e Q k f (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) f = (cid:18)Z R d χ [0 ,N ] × [0 , d − ( x ) dx (cid:19) /p = N /p . (3.10.5)By (2.10.1) applied to Γ N and ˜Γ N together with (3.10.4)and (3.10.5) we obtain N /q ≈ N /p . This forces p = q as we wanted. (cid:4) Proof of Corollary 2.10.3. Apply Theorems 2.6.1 and 2.7.2 to f = f s p ,q , u and ν α , as given in the statement of the corollary, and with η ( t ) = t /p . (cid:4) Proof of Lemma 2.10.4. Let f := f s p ,q to simplify notation. Since k e Q k f = | Q | − s /d +1 /p − / = | Q | − γ/d +(1 − α ) /τ − / for s = P Q ∈D s Q e Q we have k s k τℓ τ,τ ( u ,ν α ) = (cid:13)(cid:13)(cid:13) {k s Q e Q k f } Q ∈D (cid:13)(cid:13)(cid:13) τℓ τ,τ ( ν α ) = X Q ∈D k s Q e Q k τf | Q | α = X Q ∈D ( | s Q | | Q | − γ/d +(1 − α ) /τ − / | Q | α/τ ) τ = X Q ∈D ( | s Q | | Q | − γ/d +1 /τ − / ) τ = k s k τb γτ,τ . (cid:4) Application to real interpolation (proofs).Proof of Theorem 2.11.1. Write ξ = 1 /τ − /p > α = 1 suchthat γ = s + (1 − α ) ξd ( i.e. α = 1 − γ − sξd ), which is possible since γ = s . Once α is chosen, take s ∈ R in such a way that α = p ( s − sd ) ( i.e. s = s + αdp ). Theorem2.10.2 shows that f sp,q satisfies 1) of Theorems 2.6.1 and 2.7.2 with η ( t ) = t /p , forthe ”normalization” space f := f s p,p and ν α (notice that α = p ( s − sd ) is the conditionrequired in Lemma 2.10.4). Thus, the space ℓ τ,µ ( u , ν α ) satisfies the Jackson andBernstein’s inequalities of order ξ = 1 /τ − /p > 0, where u = {k e Q k f } Q ∈D . Taking µ = τ , Lemma 2.10.4 shows that b γτ,τ satisfies the Jackson and Bernstein’s inequalitiesof order ξ = 1 /τ − /p > 0, since γ = s + (1 − α ) ξd (the required condition).By Theorem 2.8.1, for 0 < θ < f sp,q , b γτ,τ ) θ,τ θ = A θξτ θ ( f sp,q , ν α ) . Since τ θ = (1 − θ ) p + θτ = θ ( τ − p ) + p = θξ + p , Corollary 2.10.6 gives A θξτ θ ( f sp,q , ν α ) = b ˜ γτ θ ,τ θ with ˜ γ = s + dθξ (1 − α ) = s + dθξ ( γ − s ) ξd = s + θ ( γ − s ) = (1 − θ ) s + θγ , which provesthe result. (cid:4) Acknowledgements . We thank Gustavo Garrig´os for reading a first manuscriptof this work and for his suggestions to improve the presentation. References [1] A. Almeida, Wavelet bases in generalized Besov spaces , J. Math. Anal. Appl. (2005) 304:198-211.[2] J. Bergh and J. L¨ofstr¨om, Interpolation Spaces: an Introduction , Springer-Verlag, 1976.[3] C. 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Eugenio Hern´andez, Departamento de Matem´aticas, Universidad Aut´onoma de Ma-drid, 28049 Madrid, Spain E-mail address : [email protected] Daniel Vera, Departamento de Matem´aticas, Universidad Aut´onoma de Madrid,28049 Madrid, Spain E-mail address ::