Jacobi decomposition of weighted Triebel Lizorkin and Besov spaces

Transcription

1 STUDIA MATHEMATICA 186 (2) (2008) Jacobi decomosition of weighted Triebel Lizorkin and Besov saces by George Kyriazis (Nicosia), Pencho Petrushev (Columbia, SC) and Yuan Xu (Eugene, OR) Abstract. The Littlewood Paley theory is extended to weighted saces of distributions on [ 1, 1] with Jacobi weights w(t) = (1 t) α (1+t) β. Almost exonentially localized olynomial elements (needlets) {ϕ ξ }, {ψ ξ } are constructed and, in comlete analogy with the classical case on R n, it is shown that weighted Triebel Lizorkin and Besov saces can be characterized by the size of the needlet coefficients { f, ϕ ξ } in resective sequence saces. 1. Introduction. The ϕ-transform of Frazier and Jawerth [6, 7, 8] is a owerful tool for decomosition of saces of functions or distributions on R n. Our goal in this aer is to develo similar tools for decomosition of weighted saces of distributions on [ 1, 1] with Jacobi weights (1.1) w(x) := w α,β (x) := (1 x) α (1 + x) β, α, β > 1/2. We will build uon the elements constructed in [13] and termed needlets. The targeted saces are weighted Triebel Lizorkin and Besov saces on [ 1, 1]. The main vehicle in constructing our building blocks will be the classical Jacobi olynomials {P (α,β) n } n=0 L 2 (w) := L 2 ([ 1, 1], w) and are normalized by P n (α,β) articular, (1.2) 1 1 P (α,β) n, which form an orthogonal basis for (1) = ( ) n+α [18]. In (x)p m (α,β) (x)w(x) dx = δ n,m h (α,β) n, where h (α,β) n n 1 with constants of equivalence deending only on α and β. Then the normalized Jacobi olynomials P n (x) = P (α,β) n (x), defined by (1.3) P n (x) := (h (α,β) n ) 1/2 P n (α,β) (x), n = 0, 1,..., 2000 Mathematics Subject Classification: 42A38, 42B08, 42B15. Key words and hrases: localized olynomial kernels, Jacobi weight, Triebel Lizorkin saces, Besov saces, frames. The second author has been suorted by NSF Grant DMS and the third author by NSF Grant DMS [161] c Instytut Matematyczny PAN, 2008 n

2 162 G. Kyriazis et al. form an orthonormal basis for L 2 (w), where the inner roduct is defined by 1 (1.4) f, g := f(x)g(x)w(x) dx. Consequently, for every f L 2 (w), (1.5) f = a n (f)p n with a n (f) := f, P n. n=0 Then the kernel of the nth artial sum oerator is n (1.6) K n (x, y) := P ν (x)p ν (y). 1 ν=0 Our construction of needlets relies on the fundamental fact [13] that if the coefficients on the right in (1.6) are smoothed out by samling a comactly suorted C cut-off function, then the resulting kernel has nearly exonential localization around the main diagonal y = x in [ 1, 1] 2. To be more secific, let ( ) j (1.7) L n (x, y) := â P j (x)p j (y) n j=0 with â admissible in the sense of the following definition: Definition 1.1. A function â C [0, ) is said to be admissible of tye (a) if su â [0, 2] and â(t) = 1 on [0, 1], and of tye (b) if su â [1/2, 2]. As a comanion to the weight w(x) we introduce the quantity (1.8) W(n; x) = W α,β (n; x) := (1 x + n 2 ) α+1/2 (1 + x + n 2 ) β+1/2. We will also need the distance on [ 1, 1] defined by (1.9) d(x, y) := arccos x arccos y. Now one of the main results from [13] can be stated as follows: Let â be admissible. Then for any σ > 0 there is a constant c σ > 0 deending only on σ, α, β, and â such that n (1.10) L n (x, y) c σ, x, y [ 1, 1]. W(n; x) W(n; y)(1+nd(x, y)) σ The kernels L n (x, y) are the main ingredient in constructing needlet systems here. Our construction utilizes a semidiscrete Calderón tye decomosition combined with discretization using the Gaussian quadrature formula (see 3). Earlier in [11] a similar scheme has been used for the construction of frames on the shere.

3 Jacobi decomosition of weighted saces 163 Denoting by {ϕ ξ } ξ X and {ψ ξ } ξ X the constructed analysis and synthesis needlet systems, indexed by a multilevel set X = j=0 X j, we show that every distribution f on [ 1, 1] (f D ) has the reresentation f = ξ X f, ϕ ξ ψ ξ. In this article we use the needlets to characterize two scales of weighted Triebel Lizorkin (F-saces) and Besov saces (B-saces) on [ 1, 1] defined via Jacobi exansions. The idea of using orthogonal or sectral decomositions for introducing Triebel Lizorkin and Besov saces is natural and well known (see [15, 19]). To be more recise, let ( ) ν Φ 0 (x, y) := P 0 (x)p 0 (y), Φ j (x, y) := â P ν (x)p ν (y), j 1, ν=0 2 j 1 where â is admissible of tye (b) (see Definition 1.1) and â > 0 on [3/5, 5/3]. The first scale of F-saces F with s R, 0 < <, 0 < q, is defined ( 4) as the sace of all distributions f on [ 1, 1] such that ( f F := (2 sj Φ j f( ) ) q) 1/q L <. (w) j=0 We define a second scale of F -saces F that f F := ( 5) as the sace of all f D such ( [2 sj W(2 j ; ) s Φ j f( ) ] q) 1/q L <. (w) j=0 (For the definition of Φ j f, see (2.32).) The corresonding scales of weighted Besov saces B (see [16, 19]) and B with s R, 0 <, q, are defined ( 6 7) via the (quasi-)norms ( f B := (2 sj Φ j f L (w)) q) 1/q and f B j=0 ( := [2 sj W(2 j ; ) s Φ j f( ) L (w)] q) 1/q. j=0 To some extent the second scales of F- and B-saces are more natural than the first since they embed correctly with resect to the smoothness arameter s (see 5, 7 for details). Also, the second scale of B-saces rovides the smoothness saces of nonlinear n-term aroximation from needlets ( 8). One of our main results ( 4) asserts that for all indices the weighted Triebel Lizorkin saces F can be characterized in terms of the size of the

4 164 G. Kyriazis et al. needlet coefficients, namely, ( f F 2 ) 1/q sjq f, ϕ ξ ψ ξ ( ) q L(w). j=0 ξ X j The needlet characterization of the Besov saces B ( 6) takes the form ( f B 2 sjq[ ] q/ ) 1/q. f, ϕ ξ ψ ξ j=0 ξ X j Characterizations of similar nature are obtained for the second scales of weighted Triebel Lizorkin and Besov saces F and B (see 5, 7). Using L (w) multiliers we show that the sace F 02 = F 02 can be identified as L (w) for 1 < <. This is a follow-u aer of [13]; it is closely related to [11] and [9], where needlet decomositions of Triebel Lizorkin and Besov saces on the unit shere and ball are develoed. The rest of the aer is organized as follows. In 2, some auxiliary facts are given, including localized and reroducing olynomial kernels, Gaussian quadrature, the maximal inequality, and basics of distributions on [ 1, 1]. In 3, we construct the needlets and show some of their roerties. The first and second scales of weighted Triebel Lizorkin saces are defined and characterized via neadlets in 4 and 5, resectively, while the first and second scales of Besov saces are defined and characterized via needlets in 6 and 7. In 8, Besov saces are alied to weighted nonlinear aroximation from needlets; a Jackson theorem is roved. Section 9 is an aendix, where the roofs of some statements are given. Throughout the aer we use the following notation: ( 1 1/, f := f(x) w(x) dx) 0 < <, f := su f(x). x [ 1,1] 1 For a measurable set E [ 1, 1], we set µ(e) := E w(y) dy; 1 E is the characteristic function of E and 1E := µ(e) 1/2 1 E is the L 2 (w) normalized characteristic function of E. Also, Π n denotes the set of all univariate algebraic olynomials of degree n. Positive constants are denoted by c, c 1, c,... and they may vary at every occurrence. The notation A B means c 1 A B c 2 A. 2. Preliminaries 2.1. Localized kernels induced by Jacobi olynomials. To a large extent our develoment in this aer relies on the nearly exonential localization (1.10) of kernels L n (x, y) of the form (1.7) with admissible â, established in [13]. To avoid some otential confusion, we note that the

5 Jacobi decomosition of weighted saces inner roduct in [13] is defined by f, g := c α,β 1 f(x)g(x)w(x) dx with c 1 α,β := 1 1 w(x) dx and as a result L n(x, y) from (1.7) is a constant multile of L α,β (x, y) from [13]. A similar remark alies to the constants h (α,β) n from (1.2) and [13]. The roof of estimate (1.10) (see [13]) is based on the almost exonential localization of the univariate olynomial: (2.1) L α,β n (x) := ( ) j â n j=0 (h (α,β) j ) 1 P (α,β) j (1)P (α,β) j Theorem 2.1 ([1, 13]). Assume that α β > 1/2 and let â be admissible. Then for every k 1 there exists a constant c k > 0 deending only on k, α, β, and â such that (2.2) L α,β n 2α+2 n (cos θ) c k, 0 θ π. (1 + nθ) k+α β The deendence of c k on â is of the form c k = c(α, β, k) max 1 ν k â (ν) L 1. This estimate was roved in [13] with â admissible of tye (b) and in [1] with â admissible of tye (a) (for a roof, see also [14]). In [13, Proosition 1] it is shown that (1.10) yields the following uer bound for the weighted L integrals of L n (x, y) : (2.3) 1 1 ( L n (x, y) w(y) dy c n W(n; x) (x). ) 1, 1 x 1, 0 < <. The next theorem shows that in a sense the kernel L n (x, y) from (1.7) is Li 1 in x (and y). Theorem 2.2. Let α, β > 1/2. Suose â is admissible and σ > 0 is an arbitrary constant. If x, y, z, ξ [ 1, 1], d(x, ξ) c n 1 and d(z, ξ) c n 1 with n 1, c > 0, then (2.4) L n (x, y) L n (ξ, y) c σ n 2 d(x, ξ) W(n; y) W(n; z)(1 + nd(y, z)) σ, where c σ > 0 deends only on σ, α, β, c, and â. The roof of this theorem is given in the aendix. Lower bound estimates for the integrals of L n (x, y) are nontrivial and will be vital for our further develoment. Proosition 2.3. Let â be admissible and â(t) c > 0 for t [3/5, 5/3]. Then 1 (2.5) L n (x, y) 2 w(y) dy cnw(n; x) 1, 1 x 1. 1

6 166 G. Kyriazis et al. Proof. By the definition of L n (x, y) in (1.7) and the orthogonality of the Jacobi olynomials, it follows that 1 2n L n (x, y) 2 w(y) dy c â(k/n) 2 [P (α,β) k (x)] 2. 1 k=[n/2] Since â(t) c > 0 for t [3/5, 5/3] and P k (x) = (h (α,β) k ) 1/2 P (α,β) k (x) k 1/2 P (α,β) k (x), it suffices to rove that [5n/3] k=[3n/5] [P (α,β) k (x)] 2 cw(n; x) 1, c > 0, which is established in the following roosition. Proosition 2.4. If α, β > 1 and ε > 0, then (2.6) Λ n (x) := n+[εn] k=n where c > 0 deending only on α, β, and ε. [P (α,β) k (x)] 2 cw(n; x) 1, x [ 1, 1], n 1/ε, This roosition is nontrivial and its roof is given in the aendix Reroducing kernels and best olynomial aroximation. We let E n (f) denote the best aroximation of f L (w) from Π n, i.e. (2.7) E n (f) := inf g Π n f g. To simlify our notation we introduce the following convolution : For functions Φ : [ 1, 1] 2 C and f : [ 1, 1] C, we write 1 (2.8) Φ f(x) := Φ(x, y)f(y)w(y) dy. 1 Lemma 2.5. Suose â is admissible of tye (a) and let L n (x, y) be the kernel defined in (1.7). (i) L n (x, y) is a symmetric reroducing kernel for Π n, i.e. L n g = g for g Π n. (ii) For any f L (w), 1, we have L n f Π 2n, (2.9) L n f c f and f L n f ce n (f). Proof. Part (i) is immediate since â(ν/n) = 1 for 0 ν n. The lefthand estimate in (2.9) follows from (2.3) when = 1 and = ; the general case follows by interolation. The right-hand estimate in (2.9) follows from the left-hand estimate and (i).

7 Jacobi decomosition of weighted saces 167 Lemma 2.5(i) and (2.3) are instrumental in roving Nikolski tye inequalities. Proosition 2.6. For 0 < q and g Π n, (2.10) g cn (2+2 min{0,max{α,β}})(1/q 1/) g q, furthermore, for any s R, (2.11) W(n; ) s g( ) cn 1/q 1/ W(n; ) s+1/ 1/q g( ) q. The roof of this roosition is given in the aendix Quadrature formula and subdivision of [ 1, 1]. For the construction of our building blocks (needlets) we will utilize an aroriate Gaussian quadrature formula. Let ξ j,ν =: cos θ ν, ν = 1, 2,..., 2 j+1, be the zeros of the Jacobi olynomials P (α,β) 2 j+1 ordered so that 0 < θ 1 < < θ 2 j+1 < π. It is well known that uniformly (see [5] and also (9.9) (9.10) below) (2.12) θ ν+1 θ ν 2 j and hence θ ν ν2 j. Define now (2.13) X j := {ξ j,ν : ν = 1, 2,..., 2 j+1 }, j 0, X := X j. As is well known [18] the zeros of the Jacobi olynomial P (α,β) serve as 2 j+1 knots of the Gaussian quadrature 1 (2.14) f(x)w(x) dx c ξ f(ξ), 1 ξ X j which is exact for all olynomials of degree at most 2 j+2 1. Furthermore, the coefficients c ξ are ositive and have the asymtotics (2.15) c ξ λ 2 j+1(ξ) 2 j w(ξ)(1 ξ 2 ) 1/2 2 j W(2 j ; ξ), where λ 2 j+1(t) is the Christoffel function and the constants of equivalence deend only on α, β (cf. e.g. [12]). We next introduce the jth level weighted dyadic intervals. Set as above ξ j,ν =: cos θ ν and define (2.16) (2.17) I ξj,ν := [(ξ j,ν+1 + ξ j,ν )/2, (ξ j,ν 1 + ξ j,ν )/2], ν = 2, 3,..., 2 j+1 1, I ξj,1 := [(ξ j,2 + ξ j,1 )/2, 1], I ξj,2 j+1 := [ 1, (ξ j,2 j+1 + ξ j,2 j+1 1)/2]. For ξ X j we will briefly write I ξ := I ξj,ν if ξ = ξ j,ν. It follows by (2.12) that there exist constants c 1, c 2 > 0 such that (2.18) B ξ (c 1 2 j ) I ξ B ξ (c 2 2 j ), j=0

8 168 G. Kyriazis et al. where B y (r) := {x [ 1, 1] : d(x, y) r} with d(, ) being the distance from (1.9). Also, it is straightforward to show that (2.19) µ(i ξ ) := w(x) dx 2 j W(2 j ; ξ) c ξ, ξ X j, j 0. I ξ (2.20) (2.21) It will be useful to note that W(n; cos θ) (sin θ + n 1 ) 2α+1, 0 θ 2π/3, W(n; cos θ) (sin θ + n 1 ) 2β+1, π/3 θ π. The following simle inequality will be instrumental in various roofs: (2.22) W(n; x) cw(n; y)(1+nd(x, y)) 2 max{α,β}+1, x, y [ 1, 1], n 1. For the roof see the aendix The maximal inequality. For every 0 < t < and x [ 1, 1], we define ( 1 1/t (2.23) M t f(x) := su f(y) t w(y) dy), I x µ(i) where the su is over all intervals I [ 1, 1] containing x. It is not hard to see that µ is a doubling measure on [ 1, 1] and hence the general theory of maximal inequalities alies. In articular the Fefferman Stein vectorvalued maximal inequality holds (see [17]): If 0 < <, 0 < q and 0 < t < min{, q} then for any sequence of functions {f ν } ν=1 on [ 1, 1], ( (2.24) M t f ν ( ) q) 1/q ( c f ν ( ) q) 1/q. ν=1 We need to estimate (M t 1 Iξ )(x) for the intervals I ξ from (2.16) (2.17) and other intervals. Lemma 2.7. Let η [0, 1] and 0 < ε π. Then for x [ 1, 1], ( ) 1/t ( (2.25) (M t 1 Bη(ε))(x) and hence (2.26) c ( 1 + d(η, x) ε I ν=1 d(η, x) ε + d(η, 1) ) d(η, x) (2α+2)/t (M t 1 ε Bη(ε))(x) c( 1 + Here the constants deend only on α, β, and t. ) (2α+1)/t ) d(η, x) 1/t. ε A similar lemma holds for η [ 1, 0). We relegate the roof of this lemma to the aendix.

9 Jacobi decomosition of weighted saces Distributions on [ 1, 1]. Here we give some basic and well known facts about distributions on [ 1, 1]. We will use as test functions the set D := C [ 1, 1] of all infinitely differentiable comlex-valued functions on [ 1, 1], where the toology is induced by the seminorms (2.27) φ k := φ (k) (t), k = 0, 1,.... Note that the Jacobi olynomials {P n } belong to D. More imortantly, the sace D of test functions φ can be comletely characterized by the coefficients of their Jacobi exansions: a n (φ) := φ, P n := 1 1 φ(x)p n(x)w(x) dx. Define (2.28) N k (φ) := su (n + 1) k a n (φ). n 0 Lemma 2.8. (a) φ D if and only if a n (φ) = O(n k ) for all k. (b) For every φ D we have φ = n=0 a n(φ)p n, where the convergence is in the toology of D. (c) The toology in D can be equivalently defined by the norms N k ( ), k = 0, 1,.... Proof. If φ D, then due to the orthogonality of P n to Π n 1, we have for n = 1, 2,..., a n (φ) = φ, P n = φ Q n 1, P n E n 1 (φ) 2 c k n k φ (k), where Q n 1 Π n 1 is the olynomial of best L 2 (w) aroximation to φ. Here we used a simle Jackson estimate for aroximation from algebraic olynomials (E n (φ) c k n k φ (k) ). Therefore, a n (φ) = O(n k ) and N k (φ) c k φ k for k = 0, 1,.... On the other hand, by Markov s inequality it follows that P (k) n L [ 1,1] n 2k P n L [ 1,1] cn 2k h 1/2 n P n (1) cn 2k+α+1/2. Hence, if a n (φ) = O(n k ) for all k, then φ (k) = n=0 a n(φ)p (k) n with the series converging uniformly and φ k c a n (φ) (n + 1) 2k+α+1/2 cn 2k+[α+1/2]+1 (φ), k = 0, 1,..., n=0 which comletes the roof of the lemma. The sace D := D [ 1, 1] of distributions on [ 1, 1] is defined as the set of all continuous linear functionals on D. The airing of f D and φ D will be denoted by f, φ := f(φ), which will be shown to be consistent with the inner roduct f, g := 1 1 f(x)g(x)w(x) dx in L2 (w). We will need the reresentation of distributions from D in terms of Jacobi olynomials.

10 170 G. Kyriazis et al. Lemma 2.9. (a) A linear functional f on D is a distribution (f D ) if and only if there exists k 0 such that (2.29) f, φ c k N k (φ) for all φ D, For f D, denote a n (f) := f, P n. Then for some k 0, (2.30) f, P n c k (n + 1) k, n = 0, 1,.... (b) Every f D has the reresentation f = n=0 a n(f)p n in distributional sense, i.e. (2.31) f, φ = a n (f) P n, φ = a n (f)a n (φ) for all φ D, n=0 where the series converges absolutely. Proof. (a) Part (a) follows immediately from the fact that the toology in D can be defined by the norms N k ( ) defined in (2.28). (b) Using Lemma 2.8(b) we get, for φ D, f, φ = lim f, N N a n (φ)p n = lim n=0 n=0 N n=0 N a n (φ) f, P n = a n (f)a n (φ), where for the last equality we used (2.30) and the fact that a n (φ) are raidly decaying. It is convenient to extend the convolution from (2.8) to the case of distributions. Definition Assuming that f D and Φ : [ 1, 1] 2 C is such that Φ(x, y) belongs to D as a function of y (Φ(x, ) D), we define Φ f by (2.32) Φ f(x) := f, Φ(x, ), where on the right f acts on Φ(x, y) as a function of y. 3. Construction of building blocks (needlets). Following the ideas from [13] we next construct two sequences of comanion analysis and synthesis needlets. Our construction is based on a Calderón tye reroducing formula. Let â, b satisfy the conditions (3.1) (3.2) (3.3) Hence, (3.4) â, b C [0, ), su â, b [1/2, 2], â(t), b(t) > c > 0 if t [3/5, 5/3], â(t) b(t) + â(2t) b(2t) = 1 if t [1/2, 1]. â(2 ν t) b(2 ν t) = 1, t [1, ). ν=0 n=0

11 Jacobi decomosition of weighted saces 171 It is easy to show that if â satisfies (3.1) (3.2), then there exists b satisfying (3.1) (3.2) such that (3.3) holds true (see e.g. [7]). Assuming that â, b satisfy (3.1) (3.3), we define Φ 0 (x, y) = Ψ 0 (x, y) := P 0 (x)p 0 (y), (3.5) (3.6) Φ j (x, y) := Ψ j (x, y) := ν=0 ν=0 ( ν â 2 j 1 ) P ν (x)p ν (y), j 1, ( ) ν b 2 j 1 P ν (x)p ν (y), j 1. Let X j be the set of knots of the quadrature formula (2.14), defined in (2.13), and let c ξ be the coefficients of the same quadrature. We define the jth level needlets by (3.7) ϕ ξ (x) := c 1/2 ξ Φ j (x, ξ) and ψ ξ (x) := c 1/2 ξ Ψ j (x, ξ), ξ X j. As in (2.13) we write X := j=0 X j, where equal oints from different levels X j are considered as distinct elements of X, so that X can be used as an index set. We define the analysis and synthesis needlet systems Φ and Ψ by (3.8) Φ := {ϕ ξ } ξ X, Ψ := {ψ ξ } ξ X. By estimate (1.10) it follows that the needlets have nearly exonential localization, namely, for x [ 1, 1], (3.9) Φ j (ξ, x), Ψ j (ξ, x) c σ 2 j W(2 j ; ξ) W(2 j ; x)(1 + 2 j d(ξ, x)) σ σ, and hence (3.10) ϕ ξ (x), ψ ξ (x) c σ 2 j/2 W(2 j ; x)(1 + 2 j d(ξ, x)) σ σ. Note that x in the term W(2 j ; x) above can be relaced by ξ (uon relacing c σ by a larger constant), namely, (3.11) ϕ ξ (x), ψ ξ (x) c σ 2 j/2 W(2 j ; ξ)(1 + 2 j d(ξ, x)) σ This estimate follows from (3.10) and (2.22). We will need to estimate the norms of the needlets. We have, for 0 <, ( (3.12) ϕ ξ ψ ξ 1Iξ 2 j W(2 j ; ξ) Moreover, there exist constants c, c > 0 such that (3.13) ϕ ξ L (B ξ (c 2 j )), ψ ξ L (B ξ (c 2 j )) c ( σ. ) 1/2 1/, ξ X j. 2 j W(2 j ; ξ) ) 1/2,

12 172 G. Kyriazis et al. where B ξ (c 2 j ) := {x [ 1, 1] : d(ξ, x) c 2 j }, which is an interval. Notice that if â, b in (3.1) (3.3) are real-valued then by Proosition 2.4, ( 2 j ) 1/2 (3.14) ϕ ξ (ξ), ψ ξ (ξ) c W(2 j, c > 0. ; ξ) For the roofs of (3.12) (3.13), see the aendix. Our next goal is to establish needlet decomositions of D and L (w). Proosition 3.1. (a) For f D, we have (3.15) f = Ψ j Φ j f in D, (3.16) j=0 f = ξ X f, ϕ ξ ψ ξ in D. (b) If f L (w), 1, then (3.15) (3.16) hold in L (w). Moreover, if 1 < <, then the convergence in (3.15) (3.16) is unconditional. Proof. (a) Let f D. By (2.32) and Lemma 2.9, we have 2 j ( ) ν (3.17) Φ j f = â a ν (f)p ν ν=0 ν=0 2 j 1 and further 2 j ( ) ( ) ν ν (3.18) Ψ j Φ j f = â b 2 j 1 2 j 1 a ν (f)p ν. Now (3.15) follows from (3.4) and Lemmas Note that Ψ j (x, y) and Φ j (x, y) are symmetric functions (e.g. Ψ j (y, x) = Ψ j (x, y)) and hence Ψ j Φ j (x, y) is well defined. Also, Ψ j (Φ j f) = (Ψ j Φ j ) f. We observe that Ψ j (x, u)φ j (y, u) belongs to Π 2 j+1 1 as a function of u and aly the quadrature formula from (2.14) to obtain 1 Ψ j Φ j (x, y) = Ψ j (x, u)φ j (y, u)w(u) du Hence, 1 = ξ X j c ξ Ψ j (x, ξ)φ j (y, ξ) = Ψ j Φ j f = ξ X j f, ϕ ξ ψ ξ. ξ X j ψ ξ (x)ϕ ξ (y). Substituting this in (3.15) yields (3.16). (b) To rove (3.15) in L (w) we observe that l j=0 Ψ j Φ j f = L l f with L l := l j=0 Ψ j Φ j. Because of (3.4), L l (x, y) is a reroducing kernel

13 Jacobi decomosition of weighted saces 173 for olynomials exactly as the kernels L n (x, y) from Lemma 2.5. Hence, l j=0 Ψ j Φ j f f in L (w) (1 ). Then (3.16) in L (w) follows as above. The unconditional convergence in L (w), 1 < <, follows from Proosition 4.11 and Theorem 4.5 below. Remark 3.2. It is easy to see that there exists a function â 0 satisfying (3.1) (3.2) such that â 2 (t) + â 2 (2t) = 1, t [1/2, 1]. Suose that in the above construction b = â and â 0. Then Φ j = Ψ j and ϕ ξ = ψ ξ. Now (3.16) becomes f = ξ X f, ψ ξ ψ ξ. It is easily seen that {ψ ξ : ξ X } is a tight frame for L 2 (w) (see [13]). 4. First scale of weighted Triebel Lizorkin saces on [ 1, 1]. In analogy to the classical case on R d we will define our first scale of weighted Triebel Lizorkin saces by means of the Littlewood Paley exressions emloying the kernels Φ j, defined by (4.1) Φ 0 (x, y) := P 0 (x)p 0 (y), ( ) ν Φ j (x, y) := â P ν (x)p ν (y), j 1, ν=0 where â satisfies the conditions (4.2) (4.3) 2 j 1 â C [0, ), su â [1/2, 2], â(t) > c > 0 if t [3/5, 5/3]. Definition 4.1. Let s R, 0 < <, and 0 < q. Then the weighted Triebel Lizorkin sace F := F (w) is defined as the set of all f D such that ( (4.4) f F := (2 sj Φ j f( ) ) q) 1/q < j=0 with the usual modification when q =. Observe that the above definition is indeendent of the choice of â as long as it satisfies (4.2) (4.3) (see Theorem 4.5 below). Proosition 4.2. For every s R, 0 < <, and 0 < q, F is a quasi-banach sace which is continuously embedded in D. Proof. We will only rove the continuous embedding of F in D. Then the comleteness follows by a standard argument (see e.g. [19,. 49]). Suose the kernels Φ j are as in the definition of F with â satisfying (4.2) (4.3) which are the same as (3.1) (3.2). Then as was already mentioned, there is a function b satisfying (3.1) (3.2) such that (3.3) holds as well. Let Ψ j be defined by (3.6). Then by Proosition 3.1 any function

14 174 G. Kyriazis et al. f F has the reresentation f = j=0 Ψ j Φ j f in D. Hence for φ D we have f, φ = j=0 Ψ j Φ j f, φ. Using (3.17) (3.18) we find that Ψ j Φ j f, φ = 2 j ν=2 j 2 +1 ( ν â 2 j 1 ) ( ν b 2 j 1 and alying the Cauchy Schwarz inequality yields (4.5) ) a ν (f)a ν (φ) (j 2) Ψ j Φ j f, φ c 2 j/2 Φ j f 2 max a ν(φ) 2 j 2 <ν 2 j c 2 j(2/+1/2) Φ j f max a ν(φ) 2 j 2 <ν 2 j c 2 j f F N k(φ), where k 2/ + 3/2 s, N k ( ) is from (2.28), and we used inequality (2.10). Consequently, f, φ c f F N k(φ), which is the claimed embedding. Associated to F is the sequence sace f defined as follows. Definition 4.3. Let s R, 0 < <, and 0 < q. Then f is defined as the sace of all comlex-valued sequences h := {h ξ } ξ X such that (4.6) h f := ( 2 ) 1/q sjq ( h ξ 1Iξ ( )) q < j=0 ξ X j with the usual modification for q =. Recall that 1Iξ := µ(i ξ ) 1/2 1 Iξ. We now introduce the analysis and synthesis oerators (4.7) S ϕ : f { f, ϕ ξ } ξ X and T ψ : {h ξ } ξ X ξ X h ξ ψ ξ. The next lemma shows that the oerator T ψ is well defined on f. Lemma 4.4. Let s R, 0 < <, and 0 < q. Then for any h f, T ψ h := ξ X h ξψ ξ converges in D. Moreover, the oerator T ψ : D is continuous, that is, f (4.8) T ψ h, φ cn k (φ) h f for all h f, φ D. Proof. Let h f. Then by the definition of f it follows that (4.9) 2 js h ξ 1Iξ h f, ξ X j, j 0. By (2.19), µ(i ξ ) 2 j W(2 j ; ξ) and obviously 2 (2α+2β+1)j W(2 j ; ξ) 2 2α+2β+1, which imlies 1Iξ 1 = µ(i ξ ) 1/2 1/ c2 j(2α+2β) 1/2 1/. Combining this with (4.9) we get (4.10) h ξ c2 jγ 1 h f, ξ X j, γ 1 := (2α + 2β) 1/2 1/ s.

15 Jacobi decomosition of weighted saces 175 On the other hand, for a given φ D, by Lemma 2.8, φ = n=0 φ, P n P n in D, and using the definition of ψ ξ in (3.6) (3.7) we have ψ ξ, φ = c 1/2 ξ 2 j ν=2 j 2 +1 ( ν b 2 j 1 ) P ν, φ P ν (ξ) (j 2). We use this and the rough estimates P ν cν α+β+1/2 and c ξ c to obtain ψ ξ, φ c2 jγ 2 2 j ν=2 j 2 +1 P ν, φ, γ 2 := α + β + 1/2. Combining this with (4.10) we get (4.11) h ξ ψ ξ, φ h ξ ψ ξ, φ ξ X j=0 ξ X j c h f 2 j(γ 1+γ 2 ) (#X j ) P ν, φ j=1 2 j 2 ν 2 j c h f 2 2j (ν + 1) γ1+γ2+3 P ν, φ j=1 2 j 2 ν 2 j c h f N k(φ) 2 j c h f N k(φ) <, where k := [γ 1 + γ 2 ] + 4 and for convenience P 1/2 := P 0. Therefore, the above series converges and hence the series ξ X h ξψ ξ converges in D. We define T ψ h by T ψ h, φ := ξ X h ξ ψ ξ, φ for all φ D. We finally note that estimate (4.8) is immediate from (4.11). j=1 Here is our main result concerning the weighted F-saces. Theorem 4.5. Let s R, 0 < < and 0 < q. Then the oerators S ϕ : F f and T ψ : f F are bounded and T ψ S ϕ = Id on F. Consequently, for f D we have f F if and only if { f, ϕ ξ } ξ X f. Furthermore, ( f F { f, ϕ ξ } f 2 ) 1/q sjq f, ϕ ξ ψ ξ ( ) q (4.12). j=0 ξ X j is indeendent of the selection of â satis- In addition, the definition of F fying (4.2) (4.3). For the roof of this theorem we will need several lemmas whose roofs are given in the aendix.

16 176 G. Kyriazis et al. Lemma 4.6. If ξ X j, j 0, and 0 < t <, then (4.13) ϕ ξ (x), ψ ξ (x) c(m t 1Iξ )(x), x [ 1, 1], (4.14) 1Iξ (x) c(m t ϕ ξ )(x), c(m t ψ ξ )(x), x [ 1, 1]. Lemma 4.7. For any σ > 0 there exists a constant c σ > 0 such that (4.15) Φ j ψ ξ (x) c σ 2 j/2 W(2 j ; x)(1 + 2 j d(ξ, x)) σ, ξ X ν, j 1 ν j + 1, and Φ j ψ ξ (x) = 0 for ξ X ν, ν j +2 or ν j 2. Here X ν := if ν < 0. Definition 4.8. For a collection of comlex numbers {h ξ } ξ Xj we let (4.16) h ξ := h η (1 + 2 j d(η, ξ)) σ. η X j Here σ > 1 is sufficiently large and will be selected later on. Lemma 4.9. Suose that P Π 2 j, j 0, and let a ξ := max x Iξ P (x). There exists r 1, deending only on σ, α, and β, such that if then b ξ := max{min x I η P (x) : η X j+r, I ξ I η }, (4.17) a ξ b ξ with constants of equivalence indeendent of P, j and ξ. Lemma Assume t > 0 and let {b ξ } ξ Xj (j 0) be a collection of comlex numbers. Suose that σ > (4 max{α, β}+3)/t+1 in the definition (4.16) of b ξ. Then ( ) b ξ 1 I ξ (x) cm t b η 1 Iη (x), x I ξ, ξ X j. η X j Proof of Theorem 4.5. Suose α β. Fix 0 < t < min{, q} and let σ > (4α+3)/t+1. We first note that the right-hand side equivalence in (4.12) follows immediately from Lemma 4.6 and the maximal inequality (2.24). Assume that {Φ j } are from the definition of weighted Triebel Lizorkin saces, i.e. Φ j are defined by (4.1), where â satisfies (4.2) (4.3), the same as (3.1) (3.2). As already mentioned, there exists a function b satisfying (3.1) (3.2) such that (3.3) holds. Let Ψ j be defined by (3.6) using this b. Also, let {ϕ ξ } ξ X and {ψ ξ } ξ X be the associated needlet systems defined as in (3.7). Further, let { Φ j } be a second sequence of kernels like the kernels {Φ j } above but defined by a different function â. Also, we assume that a se-

17 Jacobi decomosition of weighted saces 177 quence of comanion kernels { Ψ j } is constructed as above and let { ϕ ξ }, { ψ ξ } be the associated needlet systems, defined as in (3.5) (3.7). So, we have two totally different systems of kernels and associated needlet systems. We first establish the boundedness of T eψ : f F, where the sace F is defined by {Φ j }. Let h f and define f := ξ h ξψ ξ. Using (4.15) we have, for x [ 1, 1], Φ j f(x) = h ξ Φ j ψ ξ (x) h ξ Φ j ψ ξ (x) ξ X j 1 ν j+1 ξ X µ c2 j/2 h ξ j 1 ν j+1 ξ X ν W(2 ν ; x)(1 + 2 ν d(ξ, x)). σ Fix η X j and set Y η := {ξ X j 1 X j X j+1 : I ξ I η } (X 1 := ). Notice that #Y η const and d(x, ξ) c2 j if x I ξ and ξ Y η. Hence, we have, for x I η, Φ j f(x) c2 j/2 c2 j/2 j 1 ν j+1 ω Y η X ν ξ X ν ω Y η h ω 1 ω(x) W(2 j ; ω) c h ξ 1 ω (x) W(2 ν ; ω)(1 + 2 ν d(ξ, ω)) σ ω Y η h ω 1 ω (x), where we also used (2.19). We now insert this in (4.4) and use Lemma 4.10 and the maximal inequality (2.24) to obtain (4.18) f F ( c [2 ] q ) 1/q sj h ω 1 Iω ( ) j=0 η X j ω Y η ( c [2 ] q ) 1/q sj h ξ 1 Iξ ( ) j=0 ξ X j ( c ( [M t j=0 ξ X j 2 sj h ξ 1Iξ ) ( )] q ) 1/q c {h ξ } f. For the second estimate above it was imortant that #Y η c. Therefore, the oerator T eψ : f F is bounded. We next rove the boundedness of the oerator S ϕ : F f, where we assume this time that F is defined in terms of {Φ j }. Let f F. Then Φ j f Π 2 j. For ξ X j, we set a ξ := max x I ξ Φ j f(x), b ξ := max{min x I η Φ j f(x) : η X j+r, I ξ I η }.

18 178 G. Kyriazis et al. Assuming that r above is the constant from Lemma 4.9, it follows from the same lemma that a ξ b ξ. Therefore, f, ϕ ξ = c 1/2 ξ Φ j f(ξ) cµ(i ξ ) 1/2 a ξ cµ(i ξ ) 1/2 a ξ cµ(i ξ) 1/2 b ξ. From this, taking into account that 1Iξ := µ(i ξ ) 1/2 1 Iξ, we obtain ( { f, ϕ ξ } f := 2 ) 1/q j [ f, ϕ ξ 1Iξ ( )] q (4.19) j 0 ξ X j ( c 2 ) 1/q j [b ξ 1 I ξ ( )] q j 0 ξ X j ( c 2 j[ ( ) ] q ) 1/q M t b ξ 1 Iξ ( ) j 0 ξ X j ( c 2 ) 1/q j, j 0 ξ X j b q ξ 1 I ξ ( ) where for the second inequality above we used Lemma 4.10 and for the third the maximal inequality (2.24). Let m η := min x Iη Φ j f(x) for ξ X j+r and define, for ξ X j, X j+r (ξ) := {w X j+r : I w I ξ }. Evidently, #X j+r (ξ) c, c = c(r). Hence, d(w, η) c(r)2 j r for w, η X j+r (ξ) and therefore m w m w c j+r d(w, η) cm η. Consequently, for any ξ X j and η X j+r (ξ), we have b ξ = max w Xj+r (ξ) m w cm η and hence b ξ 1 Iξ c m η 1 I η. η X j+r (ξ) Using this estimate in (4.19) we get ( { f, ϕ ξ } f c 2 j( ) q ) 1/q m η 1 I η ( ) j 0 η X j+r ( c 2 j[ ( ) ] q ) 1/q M t m η 1 Iη ( ) j 0 η X j+r ( c (2 ) q ) 1/q js m ξ 1 Iξ ( ) j 0 ξ X j ( c (2 js Φ j f ) q) 1/q. j 0 Thus the boundedness of S ϕ : F f is established.

19 Jacobi decomosition of weighted saces 179 The identity T ψ S ϕ = Id follows from Proosition 3.1. It remains to show that F is indeendent of the articular selection of â in the definition of {Φ j }. Denote for the moment by f F (Φ) the F-norm defined by {Φ j }. Then by the above roof it follows that and hence f F (Φ) c { f, ϕ ξ } f and { f, ϕ ξ } f f F (Φ) c { f, ϕ ξ } f c f F ( Φ) e. c f F (Φ) Now the desired indeendence follows by reversing the roles of {Φ j }, { Φ j }, and their comlex conjugates. It is natural to define the weighted otential sace (generalized weighted Sobolev sace) Hs := Hs (w), s > 0, 1, on [ 1, 1] as the set of all f D such that (4.20) f H s := (n + 1) s a n (f)p n ( ) <, n=0 where a n (f) := f, P n as in Lemma 2.9. In the next statement we identify certain weighted Triebel Lizorkin saces as weighted otential saces or L (w). and Proosition We have F s2 H s, s > 0, 1 < <, F 02 H 0 L (w), 1 < <, with equivalent norms. Therefore, for any f L (w), 1 < <, ( f ) 1/2. j=0 ξ X j f, ϕ ξ ψ ξ ( ) 2 One roves this roosition in a standard way using e.g. the multiliers from [3]. The roof can be carried out exactly as in the case of sherical harmonic exansions, given in [11, Proosition 4.3], and will be omitted. 5. Second scale of weighted Triebel Lizorkin saces on [ 1, 1]. We introduce our second scale of Triebel Lizorkin saces by utilizing again the kernels Φ j defined by (4.1) with â satisfying (4.2) (4.3) (comare with 4). Definition 5.1. Let s R, 0 < <, and 0 < q. Then the weighted Triebel Lizorkin sace F := F (w) is defined as the set of all

20 180 G. Kyriazis et al. f D such that (5.1) f F := ( [2 sj W(2 j ; ) s Φ j f( ) ] q) 1/q < j=0 with the usual modification when q =. Observe that the above definition is indeendent of the choice of â as long as â satisfies (4.2) (4.3) (see Theorem 5.3 below). Following in the footstes of the develoment from 4, it is easy to show that F is a comlete quasi-banach sace, which is continuously embedded in D. For the latter one roceeds as in the roof of Proosition 4.2, where in (4.5) one, in addition, uses the obvious estimate g 2 cn γ W(n; ) s g( ) 2, where γ := (2 min{α, β} + 1)s +, which is immediate from c 1 n 2 min{α,β} 1 W(n; x) c 2, x [ 1, 1]. We ski the details. The sequence sace f associated with F is now defined as follows. Definition 5.2. Let s R, 0 < <, and 0 < q. Then f is defined as the sace of all comlex-valued sequences h := {h ξ } ξ X such that ( (5.2) h f := [µ(i ξ ) s h ξ 1Iξ ( )] q) 1/q < ξ X with the usual modification when q =. To characterize the Triebel Lizorkin saces F we use again the oerators S ϕ and T ψ from (4.7). (One shows that T ψ is well defined on f in much the same way as in Lemma 4.4.) Theorem 5.3. Let s R, 0 < < and 0 < q. Then the oerators S ϕ : F f and T ψ : f F are bounded and T ψ S ϕ = Id on F. Consequently, for f D we have that f F if and only if { f, ϕ ξ } ξ X f. Furthermore, ( f F { f, ϕ ξ } f [µ(i ξ ) s f, ϕ ξ ψ ξ ( ) ] q) 1/q (5.3). is indeendent of the selection of â satis- In addition, the definition of F fying (4.2) (4.3). ξ X The roof of this theorem is similar to the roof of Theorem 4.5. The only new ingredient is the following lemma. Lemma 5.4. Let t > 0 and s R. Suose {b ξ } ξ Xj (j 0) is a collection of comlex numbers and let σ > (4 max{α, β} + 3)(1/t + s ) + 1 in the definition (4.16) of b ξ. Then ( ) (5.4) µ(i ξ ) s b ξ 1 I ξ (x) cm t µ(i η ) s b η 1 Iη (x), x I ξ, ξ X j, η X j

21 Jacobi decomosition of weighted saces 181 Proof. For ξ X j, µ(i ξ ) 2 j W(2 j ; ξ) and hence, using (2.22), µ(i ξ ) s b ξ c 2jsW(2j ; ξ) s b η (1 + 2 j d(ξ, η)) σ η X j c 2jsW(2j ; η) s b η (1 + 2 j d(ξ, η)) σ c(µ(i 1 η ) s b η ), η X j where σ 1 := σ (2 max{α, β} + 1) s > (4 max{α, β} + 3)/t + 1. Now (5.4) follows from Lemma Now the roof of Theorem 5.3 can be carried out as the roof of Theorem 4.5, using Lemma 5.4 in lace of Lemma 4.10 and selecting σ in the definitions of h ξ and a ξ, b ξ sufficiently large. We ski the further details. In a sense the saces F are more natural than the saces F from 4 since they embed correctly with resect to the smoothness index s. Proosition 5.5. Let 0 < < 1 <, 0 < q, q 1, and < s 1 < s <. Then we have the continuous embedding (5.5) F F s 1q 1 1 if s 1/ = s 1 1/ 1. The roof of this embedding result can be carried out as the roof of Proosition 4.11 in [9] (the argument is similar to the one in the classical case of R n, see e.g. [19,. 129]). We omit it. 6. First scale of weighted Besov saces on [ 1, 1]. To introduce the first scale of weighted Besov saces we use the kernels Φ j defined in (4.1) with â satisfying (4.2) (4.3) (see [16, 19]). Definition 6.1. Let s R and 0 <, q. Then the weighted Besov sace B := B (w) is defined as the set of all f D such that ( f B := (2 sj Φ j f ) q) 1/q <, j=0 where the l q -norm is relaced by the su-norm if q =. Note that as in the case of weighted Triebel Lizorkin saces the above definition is indeendent of the choice of â satisfying (4.2) (4.3) (see Theorem 6.4). Also, the Besov sace B (w) is a quasi-banach sace which is continuously embedded in D. It is natural to associate to the weighted Besov sace B the sequence sace b defined as follows. Definition 6.2. Let s R and 0 <, q. Then b := b (w) is defined to be the sace of all comlex-valued sequences h := {h ξ } ξ X such

22 182 G. Kyriazis et al. that h b ( := 2 j[ ] q/ ) 1/q (µ(i ξ ) 1/ 1/2 h ξ ) < j=0 ξ X j with the usual modification for = or q =. The analysis and synthesis oerators S ϕ and T ψ defined in (4.7) will lay a distinctive role in this section. The next lemma shows that the oerator T ψ is well defined on b. Lemma 6.3. Let s R, 0 <, q. Then for any h b, T ψ h := ξ X h ξψ ξ converges in D. Moreover, the oerator T ψ : b D is continuous. The roof of this lemma is similar to the roof of Lemma 4.4 and will be omitted. Our main result in this section is the following characterization of weighted Besov saces. Theorem 6.4. Let s R and 0 <, q. The oerators S ϕ : B b and T ψ : b B are bounded and T ψ S ϕ = Id on B. Consequently, for f D we have f B if and only if { f, ϕ ξ } ξ X b. Moreover, ( f B { f, ϕ ξ } b 2 sjq[ ] q/ ) 1/q. (6.1) f, ϕ ξ ψ ξ j=0 ξ X j is indeendent of the selection of â satis- In addition, the definition of B fying (4.2) (4.3). To rove this theorem we will need the following lemma whose roof is resented in the aendix. Lemma 6.5. For every P Π 2 j, j 0, and 0 <, ( ) 1/ (6.2) max P (x) µ(i ξ ) c P. x I ξ ξ X j Proof of Theorem 6.4. Note first that the right-hand equivalence of (6.1) follows immediately from (3.12). As in the roof of Theorem 4.5, assume that the kernels Φ j are defined by (4.1), where â satisfies (4.2) (4.3). Let b be such that (3.1) (3.3) hold and let Ψ j be defined by (3.6) using this b. Also, let {ϕ ξ } ξ X and {ψ ξ } ξ X be the associated needlet systems defined as in (3.7). Further, assume that { Φ j }, { Ψ j }, { ϕ ξ }, { ψ ξ } is a second set of kernels and needlets. We first rove the boundedness of the oerator T eψ : b B, where B is defined via {Φ j }. Let 0 < t < min{, 1} and σ (2α + 2)/t + α + 1/2.

23 Assume h b (2.22) we get Φ j f(x) c c c Jacobi decomosition of weighted saces 183 and set f := ξ X h ξψ ξ. Emloying Lemmata 2.7, 4.7, and j 1 ν j+1 j 1 ν j+1 j 1 ν j+1 j 1 ν j+1 h ξ Φ j ψ ξ (x) ξ X ν 2 j/2 h ξ ξ X ν W(2 j ; x)(1 + 2 j d(ξ, x)) σ 2 j/2 h ξ ξ X ν W(2 j ; ξ)(1 + 2 j d(ξ, x)) σ α 1/2 h ξ µ(i ξ ) 1/2 M t (1 Iξ )(x) (X 1 := ), ξ X ν where we also used the inequality σ (2α + 2)/t + α + 1/2. Using the maximal inequality (2.24) it follows that Φ j f h ξ µ(i ξ ) 1/2 M t (1 Iξ )( ) j 1 ν j+1 ξ X ν 1 c h ξ µ(i ξ ) /2 1 Iξ (x)w(x) dx j 1 ν j+1 ξ X ν 1 c h ξ µ(i ξ ) 1 /2. j 1 ν j+1 ξ X ν Multilying by 2 js and summing over j 0 we get f B c {h ξ } b. b, where is defined in terms of {Φ j }. Note first that We next rove the boundedness of the oerator S ϕ : B we assume that B f, ϕ ξ µ(i ξ ) 1/2 Φ j f(ξ), ξ X j. Since Φ j f Π 2 j, by Lemma 6.5 we obtain µ(i ξ ) 1 /2 f, ϕ ξ c µ(i ξ ) su Φ j f(x) c Φ j f, ξ X j ξ X x I ξ j which yields { f, ϕ } b c f B. The identity T ψ S ϕ = Id follows from Proosition 3.1. The indeendence of B from the articular selection of â in the definition of {Φ j } follows from the above exactly as in the Triebel Lizorkin case (see the roof of Theorem 4.5). Our next goal is to link the weighted Besov saces with best olynomial aroximation in L (w). Denote by A the aroximation sace of all

24 184 G. Kyriazis et al. functions f L (w) such that (6.3) f A := f + ( (2 sj E 2 j(f) ) q) 1/q <, j=0 where E n (f) denotes the best aroximation of f L (w) from Π n (see (2.7)). Proosition 6.6. Let s > 0, 1, and 0 < q. Then f B if and only if f A. Moreover, (6.4) f A Proof. Let f B assumtions on s,, and q the sace B f B.. It is easy and standard to show that under the is continuously embedded in L (w), i.e. f can be identified as a function in L (w) and f c f B. It is easy to construct (see e.g. [6]) a function â 0 satisfying (4.2) (4.3) such that â(t) + â(2t) = 1 for t [1/2, 1] and hence (6.5) â(2 ν t) = 1, t [1, ). ν=0 Assume that {Φ j } are defined by (4.1) with such an â. As in Proosition 3.1, it is easy to see that f = j=0 Φ j f in L (w). Hence, since Φ j f Π 2 j, (6.6) E 2 l(f) Φ j f, l 0. j=l+1 Now, a standard argument using (6.6) shows that f A c f B. To rove the estimate in the other direction, we note that Φ j f = Φ j (f Q) for Q Π 2 j 2 (j 2). Hence, as in Lemma 2.5, Φ j f c f Q. Therefore, Φ j f ce 2 j 2(f), j 2, Φ j f c f, which imlies f B c f A. Above we used the fact that the definition of B selection of â, satisfying (4.2) (4.3). is indeendent of the Remark 6.7. It is worth mentioning that E n (f) can be characterized via the weighted moduli of smoothness of Ditzian Totik [4]. Consequently, the weighted moduli of smoothness can be used for characterization of weighted Besov saces as well. 7. Second scale of weighted Besov saces on [ 1, 1]. We introduce a second scale of weighted Besov saces by using again as in 6 the kernels Φ j, defined by (4.1) with â satisfying (4.2) (4.3).

25 Jacobi decomosition of weighted saces 185 Definition 7.1. Let s R and 0 <, q. Then the weighted Besov sace B := B (w) is defined as the set of all f D such that ( f B := [2 sj W(2 j ; ) s Φ j f( ) ] q) 1/q <, j=0 where the l q -norm is relaced by the su-norm if q =. As for the other weighted Besov and Triebel Lizorkin saces considered here the above definition is indeendent of the choice of â satisfying (4.2) (4.3). Also, the Besov sace B (w) is a quasi-banach sace which is continuously embedded in D. The main advantages of the saces B over B are that, first, they embed correctly with resect to the smoothness index s, and secondly, the right smoothness saces in nonlinear n-term weighted aroximation from needles are defined in terms of saces B (see 8 below). Proosition 7.2. Let 0 < 1 <, 0 < q q 1, and < s 1 s <. Then we have the continuous embedding (7.1) B B s 1q 1 1 if s 1/ = s 1 1/ 1. This embedding result follows readily by alying inequality (2.11). We now define the sequence sace b (w) comanion to B (w). Definition 7.3. Let s R and 0 <, q. Then b := b (w) is defined to be the sace of all comlex-valued sequences h := {h ξ } ξ X such that ( h b := [ ] q/ ) 1/q (µ(i ξ ) s+1/ 1/2 h ξ ) < j=0 ξ X j with the usual modification for = or q =. For the characterization of weighted Besov saces B, we again emloy the oerators S ϕ and T ψ from (4.7). An argument similar to the roof of Lemma 4.4 shows that T ψ is well defined on b (see also Lemma 6.3). Theorem 7.4. Let s R and 0 <, q. The oerators S ϕ : B b and T ψ : b B are bounded and T ψ S ϕ = Id on B. Consequently, for f D we have f B if and only if { f, ϕ ξ } ξ X b. Moreover, (7.2) f B { f, ϕ ξ } b is indeendent of the selection of â satis- In addition, the definition of B fying (4.2) (4.3). ( [ j=0 ξ X j µ(i ξ ) s f, ϕ ξ ψ ξ ] q/ ) 1/q. The following additional lemma is needed for the roof of Theorem 7.4.

26 186 G. Kyriazis et al. Lemma 7.5. For every P Π 2 j, j 0, and 0 <, ( ) 1/ (7.3) W(2 j ; ξ) s su P (x) µ(i ξ ) c W(2 j ; )P ( ). ξ X x I ξ j The roof of this lemma is similar to the roof of Lemma 6.5, where one uses Lemma 5.4 in lace of Lemma We ski it. For the roof of Theorem 7.4, one roceeds as in the roof of Theorem 6.4, using Lemma 7.5 instead of Lemma 6.5. The roof will be omitted. 8. Alication of weighted Besov saces to nonlinear aroximation. We consider here nonlinear n-term aroximation for a needlet system {ψ η } η X with ϕ η = ψ η, defined as in (3.5) (3.8) with b = â, â 0. Then â satisfies â 2 (t) + â 2 (2t) = 1, t [1/2, 1]. Hence {ψ η } are real-valued. Denote by Σ n the nonlinear set consisting of all functions g of the form g = ξ Λ a ξ ψ ξ, where Λ X, #Λ n, and Λ is allowed to vary with g. Let σ n (f) denote the error of best L (w)-aroximation to f L (w) from Σ n : σ n (f) := inf g Σ n f g. The aroximation will take lace in L (w), 0 < <. Assume in the following that 0 < <, s > 0, and 1/τ := s + 1/. We write briefly Bτ s := Bτ sτ. By Theorem 7.4 and (3.12) it follows that ( 1/τ (8.1) f B s τ f, ψ ξ ψ ξ ) τ. ξ X The embedding of B s τ into L (w) lays an imortant role here. Proosition 8.1. If f Bτ s, then f can be identified as a function f L (w) and (8.2) f f, ψ ξ ψ ξ ( ) c f B s τ. ξ X We now state our main result in this section. Theorem 8.2 (Jackson estimate). If f Bτ s, then (8.3) σ n (f) cn s f B s τ, where c deends only on s,, and the arameters of the needlet system.

27 Jacobi decomosition of weighted saces 187 The roofs of this theorem and of Proosition 8.1 can be carried out exactly as the roofs of the resective Jackson estimate and embedding result in [9, 11] and will be omitted. It is an oen roblem to rove the Bernstein estimate comanion to (8.3): (8.4) g B s τ cn s g for g Σ n, 1 < <. This would enable one to characterize the rates (aroximation saces) of nonlinear n-term aroximation in L (w) (1 < < ) from needlet systems. 9. Proofs Proof of Proosition 2.2. We need the following integral reresentation of L n (x, y) from [13] (see (2.15)): π 1 (9.1) L n (x, y) = c α,β 0 0 where L α,β n (t) is defined by (2.1), L α,β n (t(x, y, r, ψ)) dm α,β (r, ψ), t(x, y, r, ψ) := 1 2 (1+x)(1+y)+ 1 2 (1 x)(1 y)r2 +r 1 x 2 1 y 2 cos ψ 1, the integral is against dm α,β (r, ψ) := (1 r 2 ) α β 1 r 2β+1 (sin ψ) 2β drdψ, and the constant c α,β is determined from π 1 c α,β 1 dm α,β (r, ψ) = For any u [ 1, 1] we will denote by θ u the only angle in [0, π] such that u = cos θ u. We will need the following lemma contained in the roof of Theorem 2.4 in [13]. Lemma 9.1. Let α, β > 1/2 and k 2α + 2β + 3. Then there is a constant c k > 0 deending only on k, α, and β such that for x, y [ 1, 1], π n 2α+1 dm α,β (r, ψ) (1 + n 1 t(x, y, r, ψ)) k where σ = k 2α 2β 3. c k 1 Wα,β (n; x) W α,β (n; y)(1 + n θ x θ y ) σ,

28 188 G. Kyriazis et al. Identity (9.1) yields (9.2) L n (x, y) L n (ξ, y) π 1 c L α,β n (t(x, y, r, ψ)) L α,β n (t(ξ, y, r, ψ)) dm α,β (r, ψ) 0 0 π 1 c 0 0 L α,β n ( ) L (I r,ψ ) t(x, y, r, ψ) t(ξ, y, r, ψ) dm α,β (r, ψ), where f = f and I r,ψ is the interval with end oints t(x, y, r, ψ) and t(ξ, y, r, ψ). From estimate (2.16) in [13] and Markov s inequality, for any k there exists a constant c k > 0 such that (9.3) L α,β n 2α+4 n ( ) L (I r,ψ ) c k max u I r,ψ (1 + n 1 u) k c k n 2α+4 [(1 + n 1 t(x, y, r, ψ)) k + (1 + n 1 t(ξ, y, r, ψ)) k ]. For the rest of the roof we assume that k > 0 is sufficiently large. From the definition of t(x, y, r, ψ) one easily obtains 1 t(x, y, r, ψ) = 2 sin 2 θ x θ y + 2 sin 2 θ x θ 2 2 sin2 y 2 (1 r2 ) + sin θ x sin θ y (1 r cos ψ), which imlies t(x, y, r, ψ) t(ξ, y, r, ψ) = cos(θ ξ θ y ) cos(θ x θ y ) + (cos θ ξ cos θ x ) sin 2 θ y 2 (1 r2 ) + (sin θ ξ sin θ x ) sin θ y (1 r cos ψ). It is readily seen that cos(θ ξ θ y ) cos(θ x θ y ) = 2 sin θ x + θ ξ 2θ y 2 sin θ ξ θ x 2 θ ξ θ x ( θ z θ y + cn 1 ), where we used the inequalities θ z θ ξ cn 1 and θ x θ ξ cn 1. Therefore, (9.4) t(x, y, r, ψ) t(ξ, y, r, ψ) θ ξ θ x [( θ z θ y + cn 1 ) + sin 2 θ y 2 (1 r2 ) + sin θ y (1 r cos ψ)]. We use this and (9.3) in (9.2) to obtain L n (x, y) L n (ξ, y) c θ ξ θ x (A 1 + B 1 + A 2 + B 2 + A 3 + B 3 ), where A j and B j are integrals of the same tye with A j involving t(x, y, r, ψ) and B j involving t(ξ, y, r, ψ); the indices j = 1, 2, 3 corresond to the three terms on the right-hand side of (9.4). We will estimate them searately.

29 Jacobi decomosition of weighted saces 189 Case 1. We first estimate the integral π 1 A 1 := n 2α+4 θ z θ y + cn 1 (1 + n 1 t(x, y, r, ψ)) dm α,β(r, ψ) k 0 0 as well as the integral B 1, which is the same as A 1 but with t(ξ, y, r, ψ) in lace of t(x, y, r, ψ). Using the estimate in Lemma 9.1 and the fact that θ z θ y θ x θ y + cn 1, we have n 3 ( θ z θ y + cn 1 ) A 1 c W(n; x) W(n; y)(1 + n θx θ y ) σ n 2 c W(n; x) W(n; y)(1 + n θz θ y ). σ 1 On account of (2.22) this gives the desired estimate. The integral B 1 is estimated similarly with the same bound. Case 2. We now estimate the integrals π 1 A 2 := n 2α+4 sin 2 θ y 2 (1 r2 ) (1 + n 1 t(x, y, r, ψ)) dm α,β(r, ψ) k 0 0 and B 2 which is the same but has t(ξ, y, r, ψ) in lace of t(x, y, r, ψ). By the definition of dm α,β (r, ψ), (1 r 2 )dm α,β (r, ψ) = dm α+1,β (r, ψ). Then using the estimate from Lemma 9.1 with α relaced by α + 1, we get n sin 2 y 2 A 2 c Wα+1,β (n; x) W α+1,β (n; y)(1 + n θ x θ y ) σ c n 2 Wα,β (n; x) W α,β (n; y)(1 + n θ x θ y ) σ, where we used the fact that W α+1,β (n; y) = W α,β (n; y)(sin 2 (θ y /2)+n 2 ) and hence W α+1,β (n; x) W α,β (n; x)n 2. The equivalence θ z θ y θ x θ y + cn 1 and (2.22) then give the desired estimate. The integral B 2 is estimated similarly. Case 3. We finally estimate the integrals π 1 A 3 := n 2α+4 sin θ y (1 r cos ψ) (1 + n 1 t(x, y, r, ψ)) dm α,β(r, ψ) k 0 0 and B 3 which has t(ξ, y, r, ψ) in lace of t(x, y, r, ψ). Assume first that sin θ x n 1. Using the fact that 1 t(x, y, r, ψ) sin θ x sin θ y (1 r cos ψ),

30 190 G. Kyriazis et al. we conclude that A 3 n2α+2 sin θ x π π 1 cn 2α (1 + n 1 t(x, y, r, ψ)) k 2 dm α,β(r, ψ) 1 (1 + n 1 t(x, y, r, ψ)) k 2 dm α,β(r, ψ). Now the estimate from Lemma 9.1 can be alied to get the desired estimate. Let sin θ x n 1. We have and use the fact that sin θ y sin θ y sin θ x + sin θ x θ y θ x + n 1 1 t(x, y, r, ψ) 2 sin 2 θ x θ y c(θ x θ y ) 2 2 to conclude that π 1 A 3 cn 2α+3 1 (1 + n 1 t(x, y, r, ψ)) dm α,β(r, ψ). k Alying the estimate from Lemma 9.1 we obtain the desired result. B 3 is estimated in the same way. Putting the above estimates together comletes the roof of Theorem 2.2. Proof of Proosition 2.4. Note first that it suffices to rove (2.6) only for n n 0, where n 0 is sufficiently large. This follows from the fact that P n (α,β) and P (α,β) n+1 do not have common zeros and W(n; x) 1 if n const. Furthermore, since P (α,β) k ( x) = ( 1) k P (β,α) k (x), it is sufficient to consider only the case x [0, 1]. Note that the Jacobi olynomials are normalized by P (α,β) k (1) = ( ) k+α k k α and using Markov s inequality it follows that P (α,β) k (x) ck α for 1 δk 2 x 1, where δ > 0 is a sufficiently small constant. From this one readily infers that (2.6) holds for 1 δ 1 n 2 x 1, δ 1 > 0. Define θ [0, π] from x = cos θ. Then the latter condition on x is aarently equivalent to 0 θ δ 2 n 1 with δ 2 being a ositive constant. To estimate Λ n (cos θ) for c n 1 θ π/2 with c > 0 sufficiently large, we need the following asymtotic formula of the Jacobi olynomials: For α, β > 1, ( sin θ ) α ( cos θ ) β ( ) P n (α,β) α Γ (n + α + 1) θ 1/2 (cos θ) = N J α (Nθ) 2 2 n! sin θ + θ 1/2 O(n 3/2 )

31 Jacobi decomosition of weighted saces 191 if c 0 n 1 θ π/2, where N = n + η with η := (α + β + 1)/2, J α is the Bessel function, and c 0 > 0 is an arbitrary but fixed constant (see [18, Theorem ,. 195]). Since 2/π (sin θ)/θ 1 and (cos θ)/2 1 on [0, π/2], and also Γ (n + α + 1)/n! n α, we infer from the above that ( sin θ ) 2α [P (α,β) 2 k (cos θ)] 2 c 1 [J α ((k + η)θ)] 2 c 2 k 3/2 θ 1/2 J α ((k + η)θ). Recall the well known asymtotic formula ( ) 2 1/2 J α (z) = [cos(z + γ) + O(z 1 )], z, πz where γ = απ/2 π/4. All of the above leads to ( (9.5) sin 2) θ 2α Λ n (cos θ) n+[εn] k=n (c 1 [J α ((k + η)θ)] 2 c 2 k 3/2 θ 1/2 J α ((k + η)θ) ) c n+[εn] [cos 2 (kθ + b(θ)) c (nθ) 1 ] c εn 1, nθ k=n for c 0 n 1 θ π/2, where b(θ) = ((α + β + 1)/2)θ + γ. We now use the well known identities for the Dirichlet kernel and its conjugate to obtain, for m > n, m cos 2 (kθ+b) = 1 sin(m n + 1)θ cos(n + m)θ (m n+1)+(cos 2b+sin 2b). 2 2 sin θ k=n Therefore, n+[εn] k=n ) 2 ([εn] + 1) sin θ 1 ( 2 εn 1 π ) εn εnθ 4, cos 2 (kθ + b(θ)) 1 2 ([εn] + 1) (1 whenever (2π/ε)n 1 θ π/2. Substituting this in (9.5) we obtain ( sin θ ) 2α Λ n (cos θ) c ( ) εn 2 nθ 4 c εn c ε (9.6) nθ n cε ( ) 1 θ 4 c c c ε n c θ, c > 0,