IMI Preprint Series INDUSTRIAL MATHEMATICS INSTITUTE 2006:11. Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces

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1 INDUSTRIAL MATHEMATICS INSTITUTE 26:11 Jacobi decomosition of weighted Triebel-Lizorkin and Besov saces G. Kyriazis, P. Petrushev and Y. Xu IMI Prerint Series Deartment of Mathematics University of South Carolina

2 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Abstract. The Littlewood-Paley theory is extended to weighted saces of distributions on [, 1] with Jacobi weights wt) = 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] with Jacobi weights 1.1) wx) := w α,β x) := 1 x) α 1 + x) β, α, β > /2. We will build uon the elements constructed in [12] and termed needlets. The targeted saces are weighted Triebel-Lizorkin and Besov saces on [, 1]. The main vehicle in constructing our building blocks will be the classical Jacobi olynomials {P n α,β) } n=, which form an orthogonal basis for L 2 w) := L 2 [, 1], w) and are normalized by P n α,β) 1) = ) n+α n [17]. In articular, 1.2) P α,β) n x)p m α,β) x)wx)dx = δ n,m h α,β) n, where h α,β) n n 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 ) /2 P n α,β) x), n =, 1,..., form an orthonormal basis for L 2 w), where the inner roduct is defined by 1.4) f, g := fx)gx)wx)dx. Consequently, for every f L 2 w) 1.5) f = a n f)p n with a n f) := f, P n. n= 1991 Mathematics Subject Classification. 42A38, 42B8, 42B15. Key words and hrases. Localized olynomial kernels, Jacobi weight, Triebel-LIzorkin saces, Besov saces, frames. 1

3 2 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Then the kernel of the nth artial sum oerator is n 1.6) K n x, y) := P ν x)p ν y). ν= Our construction of needlets relies on the fundamental fact [12] that if the coefficients on the right in 1.6) are smoothed out by samling a comactly suorted C function, then the resulting kernel has nearly exonential localization around the main diagonal y = x in [, 1] 2. To be more secific, let 1.7) L n x, y) := j â P j x)p j y) n) with â admissible in the sense of the following definition: Definition 1.1. A function â C [, ) is said to be admissible of tye a) if su â [, 2] and ât) = 1 on [, 1], and of tye b) if su â [1/2, 2]. As a comanion to the weight wx) we introduce the quantity 1.8) Wn; 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] defined by 1.9) dx, y) := arccos x arccos y. Now one of the main results from [12] can be stated as follows: Let â be admissible. Then for any σ > there is a constant c σ > deending only on σ, α, β, and â such that n 1.1) L n x, y) c σ, x, y [, 1]. Wn; x) Wn; y)1 + ndx, y)) σ The kernels L n x, y) are the main ingredient in constructing needlet systems here. Our construction utilizes a semi-discrete Calderón tye decomosition combined with discretization using the Gaussian quadrature formula see 3). Earlier in [1] a similar scheme has been used for the construction of frames on the shere. Denoting by {ϕ ξ } ξ X and {ψ ξ } ξ X the constructed analysis and synthesis needlet systems, indexed by a multilevel set X = X j, we show that every distribution f on [, 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-sace) and Besov saces B-saces) on [, 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 [14, 18]. To be more recise, let Φ x, y) := P x)p y) and Φ j x, y) := ν= ν ) â 2 j P ν x)p ν y), j 1, where â is admissible of tye b) see Definition 1.1) and â > on [3/5, 5/3].

4 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 3 The first scale of F -saces F with s R, < <, < q, is defined 4) as the sace of all distributions f on [, 1] such that ) 1/q L f F := 2 sj Φ j f ) ) q <. w) We define a second scale of F -saces F f F := [ 2 sj W2 j ; ) s Φ j f ) 5) as the sace of all f D such that ] ) q <. 1/q Lw) For the definition of Φ j f, see 2.32).) The corresonding scales of weighted Besov saces B see [15, 18]) and B with s R, <, q, are defined 6-7) via the quasi-)norms f B := ) q ) 1/q 2 sj Φ j f L w) and f B := [ 2 sj W2 j ; ) s Φ j f ) L w)] q ) 1/q. To some extent the second scales of F- and B-saces are more natural than the first scales since they scale 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) shows that for all indices the weighted Triebel- Lizorkin saces F can be characterized in terms of the size of the needlet coefficients, namely, f F 2 sjq ) 1/q f, ϕ ξ ψ ξ ) q ξ X j The needlet characterization of the Besov saces B f B 2 sjq[ ξ X j f, ϕ ξ ψ ξ. L w) 6) takes the form ] q/ ) 1/q. 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 2 = F 2 can be identified as L w) for 1 < <. Atomic and molecular decomosition of weighted Triebel-Lizorkin and Besov saces can be develoed using the aroach of Frazier and Jawerth [6, 7]. This enables one to make the connection between the weighted F-saces and the weighted Hardy saces on [, 1] see [2, 9]). To revent this aer from exceeding some reasonable size we leave the atomic and molecular decomositions for elsewhere. It is an oen roblem to extend the results from this article to dimensions d > 1. The missing key element is the nearly exonential localization of kernels of tye 1.7) in the multivariate case. 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]. In 3, we construct

5 4 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU 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/, f := fx) wx)dx) < <, and f := su fx). x [,1] For a measurable set E [, 1], we set µe) := E wy) dy; 1 E is the characteristic function of E and 1 E := µe) /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 1.1) of kernels L n x, y) of the form 1.7) with admissible â, established in [12]. To avoid some otential confusion, we note that the inner roduct in [12] is defined by f, g := c α,β fx)gx)wx)dx with c α,β := wx)dx and as a result L nx, y) from 1.7) is a constant multile of L α,β x, y) from [12]. A similar remark alies to the constants h α,β) n from 1.2) and [12]. The roof of estimate 1.1) see [12]) is based on the almost exonential localization of the univariate olynomial: 2.1) L α,β n x) := â j n ) h α,β) j ) P α,β) j 1)P α,β) j x). Theorem 2.1. [1, 12] Assume that α β > /2 and let â be admissible. Then for every k 1 there exists a constant c k > deending only on k, α, β, and â such that 2.2) L α,β n 2α+2 n cos θ) c k, θ π. 1 + nθ) k+α β The deendence of c k on â is of the form c k = cα, β, k) max 1 ν k â ν) L 1. This estimate was roved in [12] with â admissible of tye b) and in [1] with â admissible of tye a) for a roof, see also [13]). In [12, Proosition 1] it is shown that 1.1) yields the following uer bound for the weighted L integrals of L n x, y) : 2.3) L n x, y) n ), wy)dy c x 1, < <. Wn; x) The next theorem shows that in a sense the kernel L n x, y) from 1.7) is Li 1 in x and y).

6 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 5 Theorem 2.2. Let α, β > /2. Suose â is admissible and σ > is an arbitrary constant. If x, y, z, ξ [, 1], dx, ξ) c n and dz, ξ) c n with n 1, c >, then 2.4) L n x, y) L n ξ, y) c σ n 2 dx, ξ) Wn; y) Wn; z)1 + ndy, z)) σ, where c σ > 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 ât) c > for t [3/5, 5/3]. Then 2.5) L n x, y) 2 wy)dy cnwn; x), x 1. Proof. By the definition of L n x, y) in 1.7) and the orthogonality of the Jacobi olynomials, it follows that L n x, y) 2 wy)dy c 2n k=[n/2] âk/n) 2 [P α,β) k x)] 2. Since ât) c > for t [3/5, 5/3] and P k x) = h α,β) k k 1/2 P α,β) k x), it suffices to rove that [5n/3] k=[3n/5] [P α,β) k x)] 2 cwn; x), c >, which is established in the following roosition. Proosition 2.4. If α, β > and ε >, then 2.6) Λ n x) := n+[εn] k=n where c > deending only on α, β, and ε. ) /2 P α,β) k x) [P α,β) k x)] 2 cwn; x), x [, 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] 2 C and f : [, 1] C, we write 2.8) Φ fx) := Φx, y)fy)wy) dy. 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).

7 6 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Proof. Part i) is immediate since â ν n ) = 1 for ν n. The left-hand side estimate in 2.9) follows from 2.3) when = 1 and = ; the general case follows by interolation. The right-hand side estimate in 2.9) follows by the left-hand side estimate and i). Lemma 2.5 i) and 2.3) are instrumental in roving Nikolski tye inequalities. Proosition 2.6. For < q and g Π n, 2.1) g cn 2+2 min{,max{α,β}})1/q/) g q, furthermore, for any s R, 2.11) Wn; ) s g ) cn 1/q/ Wn; ) s+1//q g ) q. The roof of this roosition is given in the aendix Quadrature formula and subdivision of [, 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 < θ 1 < < θ 2 j+1 < π. It is well known that uniformly see [5] and also 9.9)-9.1) 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, and set X := X j. As is well known [17] the zeros of the Jacobi olynomial P α,β) 2 serve as knots of j+1 the Gaussian quadrature 2.14) fx)wx)dx ξ X j c ξ fξ), which is exact for all olynomials of degree at most 2 j+2 1. Furthermore, the coefficients c ξ are ositive and have the asymtotic 2.15) c ξ λ 2 j+1ξ) 2 j wξ)1 ξ 2 ) 1/2 2 j W2 j ; ξ), where λ 2 j+1t) is the Christoffel function and the constants of equivalence deend only on α, β cf. e.g. [11]). We next introduce the jth level weighted dyadic intervals. Set as above ξ j,ν =: cos θ ν and define 2.16) I ξj,ν := [ξ j,ν+1 + ξ j,ν )/2, ξ j,ν + ξ j,ν )/2], ν = 2, 3,..., 2 j+1 1, and 2.17) I ξj,1 := [ξ j,2 + ξ j,1 )/2, 1], I ξj,2 j+1 := [, ξ j,2 j+1 + ξ j,2 j+1 )/2]. For ξ X j we will briefly denote I ξ := I ξj,ν if ξ = ξ j,ν. It follows by 2.12) that there exist constants c 1, c 2 > such that 2.18) B ξ c 1 2 j ) I ξ B ξ c 2 2 j ),

8 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 7 where B y r) := {x [, 1] : dx, y) r} with d, ) being the distance from 1.9). Also, it is straightforward to show that 2.19) µi ξ ) := wx) dx 2 j W2 j ; ξ) c ξ, ξ X j, j. I ξ It will be useful to note that 2.2) 2.21) Wn; cos θ) sin θ + n ) 2α+1, θ 2π/3, Wn; cos θ) sin θ + n ) 2β+1, π/3 θ π. The following simle inequality will be instrumental in various roofs 2.22) Wn; x) cwn; y)1 + ndx, y)) 2 max{α,β}+1, x, y [, 1], n 1. For the roof see the aendix The maximal inequality. For every < t < and x [, 1], we define 1/t ) M t fx) := su fy) t wy) dy), I x µi) I where the su is over all intervals I [, 1] containing x. It is not hard to see that µ is a doubling measure on [, 1] and hence the general theory of maximal inequalities alies. In articular the Fefferman-Stein vector-valued maximal inequality holds see [16]): If < <, < q and < t < min{, q} then for any sequence of functions {f ν } ν=1 on [, 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. ν=1 Lemma 2.7. Let η [, 1] and < ε π. Then for x [, 1] ) /t 2.25) M t 1 Bηε))x) and hence 2.26) c 1 + dη, x) ε dη, x) ε + dη, 1) dη, x) ) 2α+2)/t Mt 1 Bη ε))x) c 1 + ε Here the constants deend only on α, β, and t. ) 2α+1)/t dη, x) ) /t. ε A similar lemma holds for η [, ). We relegate the roof of this lemma to the aendix Distributions on [, 1]. Here we give some basic and well known facts about distributions on [, 1]. We will use as test functions the set D := C [, 1] of all infinitely differentiable comlex valued functions on [, 1], where the toology is induced by the semi-norms 2.27) φ k := su φ k) t), k =, 1,.... t 1

9 8 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU 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 := φx)p nx)wx)dx. Denote 2.28) N k φ) := su n + 1) k a n φ). n Lemma 2.8. a) φ D if and only if a n φ) = On k ) for all k. b) For every φ D we have φ = n= a nφ)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 =, 1,.... Proof. If φ D, then due to the orthogonality of P n to Π n, we have for n = 1, 2,... a n φ) = φ, P n = φ Q n, P n E n φ) 2 c k n k φ k), where Q n Π n 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) = On k ) and N k φ) c k φ k for k =, 1,.... On the other hand, by Markov s inequality it follows that P k) n L [,1] n 2k P n L [,1] cn 2k h /2 n P n 1) cn 2k+α+1/2. Hence, if a n φ) = On k ) for all k, then φ k) = n= a nφ)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 =, 1,..., n= which comletes the roof of the lemma. The sace D := D [, 1] of distributions on [, 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 := fφ), which will be shown to be consistent with the inner roduct f, g := fx)gx)wx)dx in L2 w). We will need the reresentation of distributions from D in terms of Jacobi olynomials. Lemma 2.9. a) A linear functional f on D is a distribution f D ) if and only if there exists k 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 2.3) f, P n c k n + 1) k, n =, 1,.... b) Every f D has the reresentation f = n= a nf)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= where the series converges absolutely. n=

10 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 9 Proof. a) Part a) follows immediately by 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= N n= N a n φ) f, P n = a n f)a n φ), where for the last equality we used 2.3) and the fact that a n φ) are raidly decaying. It is convenient to us to extend the convolution from 2.8) to the case of distributions. Definition 2.1. Assuming that f D and Φ : [, 1] 2 C is such that Φx, y) belongs to D as a function of y Φx, ) D), we define Φ f by 2.32) Φ fx) := f, Φx, ), where on the right f acts on Φx, y) as a function of y. n= 3. Construction of building blocks Needlets) Following the ideas from [12] we next construct two sequences of comanion analysis and synthesis needlets. Our construction is based on a Calderon tye reroducing formula. Let â, b satisfy the conditions 3.1) 3.2) 3.3) Hence, 3.4) â, b C [, ), su â, b [1/2, 2], ât), bt) > c >, if t [3/5, 5/3], ât) bt) + â2t) b2t) = 1, if t [1/2, 1]. â2 ν t) b2 ν t) = 1, t [1, ). ν= 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 x, y) = Ψ x, y) := P x)p y), ν ) 3.5) Φ j x, y) := â 2 j P ν x)p ν y), j 1, 3.6) Ψ j x, y) := ν= ν= ν ) b 2 j P ν x)p ν y), j 1. Let X j be the set of knots of 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 := 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.

11 1 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU By estimate 1.1) it follows that the needlets have nearly exonential localization, namely, for x [, 1], 3.9) Φ j ξ, x), Ψ j ξ, x) c σ 2 j W2j ; ξ) W2 j ; x)1 + 2 j dξ, x)) σ σ, and hence 3.1) ϕ ξ x), ψ ξ x) c σ 2 j/2 W2j ; x)1 + 2 j dξ, x)) σ σ. Note that x in the term W2 j ; x) above can be relaced by ξ relacing c σ by a larger constant), namely, 3.11) ϕ ξ x), ψ ξ x) c σ 2 j/2 W2j ; ξ)1 + 2 j dξ, x)) σ This estimate follows by 3.1) and 2.22). We will need to estimate the norms of the needlets. We have for <, ) ϕ ξ ψ ξ 1 j ) 1/2/, Iξ ξ Xj W2 j. ; ξ) Moreover, there exist constants c, c > such that 3.13) ϕ ξ L B ξ c 2 j )), ψ ξ L B ξ c 2 j )) c 2 j σ. W2 j ; ξ) ) 1/2, where B ξ c 2 j ) := {x [, 1] : dξ, x) c 2 j } which is an interval. Notice that if â, b in 3.1)-3.3) are real valued then by Proosition j ) 1/2, 3.14) ϕ ξ ξ), ψ ξ ξ) c c >. W2 j ; 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 and 3.16) 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 2 j a ν f)p ν ν= ν= and further 2 j ν ν ) 3.18) Ψ j Φ j f = â ) b 2 j 2 j a ν f)p ν.

12 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 11 Now 3.15) follows by 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 as a function of u and aly the quadrature formula from 2.14) to obtain Hence, Ψ j Φ j x, y) = = Ψ j x, u)φ j y, u)wu)du 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 Ψ j Φ j f = L l f with L l := l Ψ j Φ j. Because of 3.4), L l x, y) is a reroducing kernel for olynomials exactly as the kernels L n x, y) from Lemma 2.5. Hence, l 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 by Proosition 4.3 and Theorem 4.5 below. Remark 3.2. It is easy to see that there exists a function â 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. 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 [12]). 4. First scale of weighted Triebel-Lizorkin saces on [, 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 ν ) 4.1) Φ x, y) := P x)p y) and Φ j x, y) := â 2 j P ν x)p ν y), j 1, where â satisfies the conditions 4.2) 4.3) ν= â C [, ), su â [1/2, 2], ât) > c >, if t [3/5, 5/3]. Definition 4.1. Let s R, < <, and < q. Then the weighted Triebel-Lizorkin sace F := F w) is defined as the set of all f D such that ) 1/q 4.4) f F := 2 sj Φ j f ) ) q < 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).

13 12 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Proosition 4.2. For every s R, < <, and < q, F 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. [18],. 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 f F has the reresentation: f = Ψ j Φ j f in D. Hence for φ D we have f, φ = Ψ j Φ j f, φ. Using 3.17)-3.18) it follows that 2 j ν ν ) Ψ j Φ j f, φ = â ) b 2 j 2 j a ν f)a ν φ) and 4.5) ν= Ψ j Φ j f, φ c2 j/2 Φ j f 2 max a νφ) ν 2 j c2 j2/+1/2) Φ j f max a νφ) ν 2 j c2 j f F N kφ), where k 2/ + 3/2 s, N k ) is from 2.28), and we used inequality 2.1). Consequently, f, φ c f F N kφ), which is the desired embedding. It is natural to define the weighted otential sace generalized weighted Sobolev sace) Hs := Hs w), s >, 1, on [, 1] as the set of all f D such that 4.6) f H s := n + 1) s a n f)p n ) <, n= 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). Proosition 4.3. We have 4.7) F s2 H s, s >, 1 < <, and 4.8) F 2 H L w), 1 < <, with equivalent norms. 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 [1, Proosition 4.3], and will be omitted. Associated with F is the sequence sace f defined as follows. Definition 4.4. Let s R, < <, and < q. Then f the sace of all comlex-valued sequences h := {h ξ } ξ X such that 4.9) h f := 2 ) 1/q sjq h ξ 1 Iξ )) q < ξ X j is a is defined as

14 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 13 with the usual modification for q =. Recall that 1 Iξ := µi ξ ) /2 1 Iξ. We now introduce the analysis and synthesis oerators 4.1) S ϕ : f { f, ϕ ξ } ξ X and T ψ : {h ξ } ξ X ξ X h ξ ψ ξ. S ϕ Here is our main result concerning the weighted F -saces. Theorem 4.5. Let s R, < < and < q. : F f and T ψ : f Consequently, for f D we have that f F Furthermore, 4.11) f F { f, ϕ ξ } f In addition, the definition of F 4.2) 4.3). Then the oerators F are bounded and T ψ S ϕ = Id on F. if and only if { f, ϕ ξ } ξ X f 2 sjq ) 1/q f, ϕ ξ ψ ξ ) q ξ X j.. is indeendent of the selection of â satisfying For the roof of this theorem we will need several lemmas whose roofs are given in the aendix. Lemma 4.6. If ξ X j, j, and < t <, then 4.12) ϕ ξ x), ψ ξ x) cm t 1 Iξ )x), x [, 1], and 4.13) 1 Iξ x) cm t ϕ ξ )x), cm t ψ ξ )x), x [, 1]. Lemma 4.7. For any σ > there exists a constant c σ > such that 4.14) Φ j ψ ξ x) c σ 2 j/2 W2j ; x)1 + 2 j dξ, x)) σ, ξ X ν, j 1 ν j + 1, and Φ j ψ ξ x) = for ξ X ν, ν j + 2 or ν j 2. Here X ν := if ν <. Definition 4.8. For a collection of comlex numbers {h ξ } ξ Xj we let 4.15) h ξ := h η j dη, 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, 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 }, 4.16) a ξ b ξ with constants of equivalence indeendent of P, j and ξ. Lemma 4.1. Assume t > and let {b ξ } ξ Xj j ) be a collection of comlex numbers. Suose that σ > 4 max{α, β} + 3)/t + 1 in the definition 4.15) of b ξ. Then ) b ξ1 Iξ x) cm t b η 1 Iη x), x I ξ, ξ X j. η X j

15 14 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Proof of Theorem 4.5. Suose α β. Fix < t < min{, q} and let σ > 4α + 3)/t + 1. We first note that the right-hand side equivalence in 4.11) 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 } from above but defined by a different function â. Also, we assume that a sequence 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 totaly different systems of kernels and associated needlet systems. We first establish the boundedness of T ψ : f F, defined similarly as in 4.1), where the sace F is defined by {Φ j }. Let h := {h ξ } ξ X be a finitely suorted sequence and f := ξ h ξψ ξ. Using 4.14) we have for x [, 1], Φ j fx) = h ξ Φ j ψ ξ x) h ξ Φ j ψ ξ x) ξ X j ν j+1 ξ X µ c2 j/2 W2ν ; x)1 + 2 ν dξ, x)). σ j ν j+1 ξ X ν h ξ Fix η X j and denote Y η := {ξ X j X j X j+1 : I ξ I η } X := ). Notice that #Y η constant and dx, ξ) c2 j if x I ξ and ξ Y η. Hence, we have for x I η Φ j fx) c2 j/2 h ξ 1 ω x) W2ν ; ω)1 + 2 ν dξ, ω)) σ c2 j/2 j ν j+1 ω Y η X ν ξ X ν ω Y η h ω1 ω x) W2j ; ω) c ω Y η h ω 1 ω x), where we also used 2.19). We now insert this in 4.4) and use Lemma 4.1 and the maximal inequality 2.24) to obtain 4.17) f F c [2 sj h 1 ] q ) 1/q ω Iω ) η X j ω Y η c [2 sj h 1 ] q ) 1/q ξ Iξ ) ξ X j c ) ] q ) 1/q [M t 2 sj h ξ 1 Iξ ) ξ X j c {h ξ } f. For the second estimate above it was imortant that #Y η c. This establishes the desired result for finitely suorted sequences. Using the continuous embedding of F in D Lemma 4.2) and the density of finitely suorted sequences in f it follows from 4.17) that for every h f, T ψh := ξ X h ψ ξ ξ is a well defined

16 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 15 distribution in D. Thus a standard density argument shows that T ψ : f is bounded. We next rove the boundedness of the oerator S ϕ : F is defined in terms of {Φ j }. Let f F this time that F For ξ X j, we set F f, where we assume. Then Φ j f Π 2 j. a ξ := max x I ξ Φ j fx), b ξ := max{min x I η Φ j fx) : η X j+r, I ξ I η }. Assuming that r above is the constant from Lemma 4.9, it follows by the same lemma that a ξ b ξ. Therefore, f, ϕ ξ = c 1/2 ξ Φ j fξ) cµi ξ ) 1/2 a ξ cµi ξ ) 1/2 a ξ cµi ξ ) 1/2 b ξ. From this, taking into account that 1 Iξ := µi ξ ) /2 1 Iξ, we obtain 4.18) { f, ϕ ξ } f := 2 ) 1/q j [ f, ϕ ξ 1 Iξ )] q j ξ X j c 2 ) 1/q j [b ξ1 Iξ )] q j ξ X j c 2 j[ ) ] q ) 1/q M t b ξ 1 Iξ ) j ξ X j c 2 ) 1/q j, j ξ X j b q ξ 1 I ξ ) where for the second inequality above we used Lemma 4.1 and for the third the maximal inequality 2.24). Let m η := min x Iη Φ j fx) for ξ X j+r and denote, for ξ X j, X j+r ξ) := {w X j+r : I w I ξ }. Evidently, #X j+r ξ) c, c = cr). Hence, dw, η) cr)2 j r for w, η X j+r ξ) and therefore m w m w c j+r dw, 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 η X j+rξ) m η1 Iη.

17 16 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Using this estimate in 4.18) we get { f, ϕ ξ } f c 2 j ) q ) 1/q m η1 Iη ) j η X j+r c 2 j[ ) ] q ) 1/q M t m η 1 Iη ) j η X j+r c 2 ) q ) 1/q js m ξ 1 Iξ ) j ξ X j c 2 js Φ j f ) q) 1/q. j Thus the boundedness of S ϕ : F f is established. The identity T ψ S ϕ = Id follows by 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 an instant 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) c f F Φ) Now the desired indeendence follows by reversing the roles of {Φ j },{ Φ j }, and their comlex conjugates. 5. Second scale of weighted Triebel-Lizorkin saces on [, 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, < <, and < q. Then the weighted Triebel-Lizorkin sace F := F w) is defined as the set of all f D such that [ ) q 1/q 5.1) f F := 2 sj W2 j ; ) s Φ j f ) ] < 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 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 embedded continuously 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 Wn; ) s g ) 2, where γ := 2 min{α, β} + 1)s +, which is immediate from c 1 n 2 min{α,β} Wn; x) c 2, x [, 1]. We ski the details. The sequence sace f associated with F is now defined as follows. Definition 5.2. Let s R, < <, and < q. Then f the sace of all comlex-valued sequences h := {h ξ } ξ X such that 5.2) h f := ξ X [ µi ξ ) s h ξ 1 Iξ )] q ) 1/q < is defined as

18 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 17 with the usual modification when q =. To characterize the Triebel-Lizorkin saces F we use again the oerators S ϕ and T ψ defined in 4.1) and the sequence saces f Theorem 5.3. Let s R, < < and < q. S ϕ : F f and T ψ : f Consequently, for f D we have that f F Furthermore, 5.3) f F { f, ϕ ξ } f In addition, the definition of F 4.2) 4.3).. Then the oerators F are bounded and T ψ S ϕ = Id on F. if and only if { f, ϕ ξ } ξ X f ξ X [ µi ξ ) s f, ϕ ξ ψ ξ ) ] q ) 1/q.. is indeendent of the selection of â satisfying 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 > and s R. Suose {b ξ } ξ Xj j ) is a collection of comlex numbers and let σ > 4 max{α, β}+3)1/t+ s )+1 in the definition 4.15) of b ξ. Then ) 5.4) µi ξ ) s b ξ1 Iξ x) cm t µi η ) s b η 1 Iη x), x I ξ, ξ X j, η X j Proof. For ξ X j, µi ξ ) 2 j W2 j ; ξ) and hence, using 2.22), µi ξ ) s b ξ c 2jsW2j ; ξ) s b η j dξ, η)) σ c 2jsW2j ; η) s b η j dξ, η)) σ 1 η X j η X j, c µi η ) s b η ) where σ 1 := σ 2 max{α, β} + 1) s > 4 max{α, β} + 3)/t + 1. Now 5.4) follows by Lemma 4.1. 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 4.1 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 scale are embedded) correctly with resect to the smoothness index s. Proosition 5.5. Let < < 1 <, < 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 similarly as in the classical case on R n using inequality 2.11) and Theorem 5.3 see e.g. [18], age 129). It will be omitted. 6. First scale of weighted Besov saces on [, 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 [15, 18]).

19 18 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Definition 6.1. Let s R and <, 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 <, 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 6.5). Also, the Besov sace B w) is a quasi-banach sace which is continuously embedded in D. Our next goal is to link the weighted Besov saces with best olynomial aroximation in L w). Recall that E n f) denotes the best aroximation of f L w) from Π n see 2.7)). Proosition 6.2. Let s >, 1, and < q. Then f B only if 6.1) f A B := f + Moreover, 6.2) f A B 2 sj E 2 j f) ) q) 1/q <. f B. if and Proof. Let f B. It is easy and standard to show that under the assumtions on s,, and q the sace B is continuously imbedded 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 satisfying 4.2)-4.3) such that ât) + â2t) = 1 for t [1/2, 1] and hence 6.3) â2 ν t) = 1, t [1, ). ν= Assume that {Φ j } are defined by 4.1) with such a function â. As in Proosition 3.1, it is easy to see that f = Φ j f in L w). Hence, since Φ j f Π 2 j, 6.4) E 2 lf) Φ j f, l. j=l+1 Now, a standard argument using 6.4) shows that f A B 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 2f), j 2, and Φ j f c f, which imly f B c f A B. Above we used that the definition of B is indeendent of the selection of â, satisfying 4.2)-4.3). Remark 6.3. Proosition 6.2 shows that when s > and 1 the weighted Besov saces B can be identified as aroximation saces induced by best olynomial aroximation in L w). Also, it is worth mentioning that E n f) can

20 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 19 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. b It is natural to associate with the weighted Besov sace B defined as follows. the sequence sace Definition 6.4. Let s R and <, 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 := 2 j[ ξ X j with the usual modification for = or q =. µi ξ ) 1//2 h ξ ) ] q/ ) 1/q < Our main result in this section is the following characterization of weighted Besov saces, which emloys the oerators S ϕ and T ψ defined in 4.1). Theorem 6.5. Let s R and <, q. The oerators S ϕ : B T ψ : b B we have that f B and are bounded and T ψ S ϕ = Id on B. Consequently, for f D if and only if { f, ϕ ξ } ξ X b. Moreover, b 6.5) f B { f, ϕ ξ } b 2 sjq[ ξ X j f, ϕ ξ ψ ξ ] q/ ) 1/q. In addition, the definition of B 4.2) 4.3). is indeendent of the selection of â satisfying To rove this theorem we will need the following lemma whose roof is resented in the aendix. Lemma 6.6. For every P Π 2 j, j, and < 6.6) ) 1/ max P x) µi ξ ) c P. x I ξ ξ X j Proof of Theorem 6.5. Note first that the right-hand side of 6.5) 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 ψ : b B, where B is defined via {Φ j }. Let < t < min{, 1} and σ 2α + 2)/t + α + 1/2. Assume that h = {h ξ } is a finitely suorted sequence and set f := ξ X h ξψ ξ. Emloying

21 2 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU Lemmata 2.7, 4.7, and 2.22) we get Φ j fx) h ξ Φ µ ψ ξ x) j ν j+1 ξ X ν 2 j/2 c h ξ W2j j ν j+1 ξ X ν ; x)1 + 2 j dξ, x)) σ 2 j/2 c h ξ W2j j ν j+1 ξ X ν ; ξ)1 + 2 j dξ, x)) σ α/2 c h ξ µi ξ ) /2 M t 1 Iξ )x), j ν j+1 ξ X ν where we also used that σ 2α + 2)/t + α + 1/2. Using the maximal inequality 2.24) it follows that Φ j f h ξ µi ξ ) /2 M t 1 Iξ ) ) j ν j+1 ξ X ν c h ξ µi ξ ) /2 1 Iξ x)wx) dx j ν j+1 ξ X ν c h ξ µi ξ ) 1 /2. j ν j+1 ξ X ν Multilying by 2 js and summing over j we get f B the result to an arbitrary sequence h = {h ξ } b Triebel-Lizorkin case by using the embedding of B suorted sequences in b. We next rove the boundedness of the oerator S ϕ : B is defined in terms of {Φ j }. Note first that that B f, ϕ ξ µi ξ ) 1/2 Φ j fξ), ξ X j. c {h ξ} b. To extend one roceeds similarly as in the in D and the density of finitely b, where we assume Since Φ j f Π 2 j, then using Lemma 6.6 µi ξ ) 1 /2 f, ϕ ξ c µi ξ ) su Φ j fx) c Φ j f, x I ξ X j ξ X ξ j which yields { f, ϕ } b c f B. The identity T ψ S ϕ = Id follows by Proosition 3.1. of the articular selection of â in the definition of {Φ j } follows from above exactly as in the Triebel-Lizorkin case see the roof of Theorem 4.5). The indeendence of B 7. Second scale of weighted Besov saces on [, 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). Definition 7.1. Let s R and <, q. Then the weighted Besov sace B := B w) is defined as the set of all f D such that f B := [ ] q ) 1/q 2 sj W2 j ; ) s Φ j f ) <,

22 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 21 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 scale are embedded) 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 < 1 <, < q q 1, and < s 1 s <. Then we have the continuous embedding 7.1) B B s1q1 1 if s 1/ = s 1 1/ 1. This embedding result follows readily by alying inequality 2.11). We now define the comanion to B w) sequence sace b w). Definition 7.3. Let s R and <, 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 := [ ξ X j with the usual modification for = or q =. µi ξ ) s+1//2 h ξ ) ] q/ ) 1/q < For the characterization of weighted Besov saces B, we again emloy the oerators S ϕ and T ψ defined in 4.1). Theorem 7.4. Let s R and <, q. The oerators S ϕ : B T ψ : b B we have that f B b and are bounded and T ψ S ϕ = Id on B. Consequently, for f D if and only if { f, ϕ ξ } ξ X b. Moreover, 7.2) f B { f, ϕ ξ } b [ ξ X j µi ξ ) s f, ϕ ξ ψ ξ ] q/ ) 1/q. In addition, the definition of B 4.2)-4.3). is indeendent of the selection of â satisfying The following additional lemma is needed for the roof of Theorem 7.4. Lemma 7.5. For every P Π 2 j, j, and < 7.3) ) 1/ W2 j ; ξ) s su P x) µi ξ ) c W2 j ; )P ). x I ξ X ξ j The roof of this lemma is similar to the roof of Lemma 6.6, where one uses Lemma 5.4 in lace of Lemma 4.1. We ski it. For the roof of Theorem 7.4, one roceeds as in the roof of Theorem 6.5, using Lemma 7.5 instead of Lemma 6.6. The roof will be omitted.

23 22 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU 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. 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), < <. Assume in the following that < <, s >, and 1/τ := s + 1/. Denote 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 B s τ, 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. The roofs of this theorem and Proosition 8.1 can be carried out exactly as the roofs of the resective Jackson estimate and embedding result in [1] and will be omitted. It is an oen roblem to rove the comanion to 8.3) Bernstein estimate: 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. Proof of Proosition 2.2. L n x, y) from [12] see 2.15)): 9.1) L n x, y) = c α,β π where L α,β n t) is defined by 2.1), 9. Aendix We need the following integral reresentation of L α,β n tx, y, r, ψ))dm α,β r, ψ), tx, y, r, ψ) := x)1 + y) x)1 y)r2 + r 1 x 2 1 y 2 cos ψ 1,

24 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 23 the integral is against dm α,β r, ψ) := 1 r 2 ) α β r 2β+1 sin ψ) 2β drdψ, and the constant c α,β is determined from c α,β π 1 dm α,β r, ψ) = 1. For any u [, 1] we will denote by θ u the only angle in [, π] such that u = cos θ u. We will need the following lemma contained in the roof of Theorem 2.4 in [12]. Lemma 9.1. Let α, β > /2 and k 2α + 2β + 3. Then there is a constant c k > deending only on k, α, and β such that for x, y [, 1] π n 2α+1 dm α,β r, ψ) 1 + n 1 tx, y, r, ψ) where σ = k 2α 2β ) Identity 9.1) yields L n x, y) L n ξ, y) c c π π 1 ) k c k Wα,β n; x) W α,β n; y)1 + n θ x θ y ), σ L α,β n tx, y, r, ψ)) L α,β n tξ, y, r, ψ)) dm α,β r, ψ) L α,β n ) L I r,ψ ) tx, y, r, ψ) tξ, y, r, ψ) dm α,β r, ψ), where f = f and I r,ψ is the interval with end oints tx, y, r, ψ) and tξ, y, r, ψ). From estimate 2.16) in [12] and Markov s inequality, for any k there exists a constant c k > such that 9.3) L α,β n ) L I r,ψ ) c k max n 2α+4 u I r,ψ 1 + n 1 u [ c k n 2α n ) k 1 tx, y, r, ψ) n ) ] k 1 tξ, y, r, ψ). For the rest of the roof we assume that k > is sufficiently large. From the definition of tx, y, r, ψ) one easily obtains 1 tx, y, r, ψ) = 2 sin 2 θ x θ y 2 which imlies ) k + 2 sin 2 θ x 2 sin2 θ y 2 1 r2 ) + sin θ x sin θ y 1 r cos ψ), tx, y, r, ψ) tξ, y, r, ψ) = cosθ ξ θ y ) cosθ 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 cosθ ξ θ y ) cosθ x θ y ) = 2 sin θ x + θ ξ 2θ y sin θ ξ θ x 2 2 θ ξ θ x θ z θ y + cn ),

25 24 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU where we used that θ z θ ξ cn and θ x θ ξ cn. Therefore, tx, y, r, ψ) tξ, y, r, ψ) θ ξ θ x [ θ z θ y + cn ) + sin 2 θ y 9.4) 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 tx, y, r, ψ) and B j involving tξ, y, r, ψ); the indices j = 1, 2, 3 corresond to the three terms in the right-hand side of 9.4). We will estimate them searately. Case 1. We first estimate the integral A 1 := n 2α+4 π θ z θ y + cn 1 + n 1 tx, y, r, ψ)) k dm α,βr, ψ) and the integral B 1, the same as A 1 but involving tξ, y, r, ψ) in lace of tx, y, r, ψ). Using the estimate in Lemma 9.1 and the fact that θ z θ y θ x θ y + cn, we have n 3 θ z θ y + cn ) A 1 c Wn; x) Wn; y)1 + n θx θ y ) σ n 2 c Wn; x) Wn; y)1 + n θz θ y ). σ 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 A 2 := n 2α+4 π sin 2 θ y 2 1 r2 ) 1 + n 1 tx, y, r, ψ)) k dm α,βr, ψ) and B 2 which is the same but involves tξ, y, r, ψ) in lace of tx, y, r, ψ). By the definition of dm α,β r, ψ) we have 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 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 θy 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 and 2.22) then give the desired estimate. The integral B 2 is estimated similarly. Case 3. We finally estimate the integrals A 3 := n 2α+4 π n 2 sin θ y 1 r cos ψ) 1 + n 1 tx, y, r, ψ)) k dm α,βr, ψ) and B 3 which involves tξ, y, r, ψ) in lace of tx, y, r, ψ). Assume first that sin θ x n. Using the fact that 1 tx, y, r, ψ) sin θ x sin θ y 1 r cos ψ),

26 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 25 we conclude that A 3 n2α+2 sin θ x π cn 2α+3 π n 1 tx, y, r, ψ)) k 2 dm α,βr, ψ) n 1 tx, y, r, ψ)) k 2 dm α,βr, ψ). Now the estimate from Lemma 9.1 can be alied to get the desired estimate. Let sin θ x n. We have and use the fact that to conclude that sin θ y sin θ y sin θ x + sin θ x θ y θ x + n 1 tx, y, r, ψ) 2 sin 2 θ x θ y 2 A 3 cn 2α+3 π cθ x θ y ) n 1 tx, y, r, ψ)) k 2 dm α,βr, ψ). Alying the estimated 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 2.4) only for n n, where n 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 Wn; x) 1 if n constant. Furthermore, since P α,β) k x) = ) k P β,α) k x), it is sufficient to consider only the case x [, 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 > is a sufficiently small constant. From this one readily infers that 2.6) holds for 1 δ 1 n 2 x 1, δ 1 >. Define θ [, π] from x = cos θ. Then the latter condition on x is aarently equivalent to θ δ 2 n with δ 2 being a ositive constant. To estimate Λ n cos θ) for c n θ π/2 with c > sufficiently large, we need the following asymtotic formula of the Jacobi olynomials: For α, β >, sin θ ) α cos θ ) β P α,β) α Γn + α + 1) θ ) 1/2Jα n cos θ) = N Nθ) 2 2 n! sin θ + θ 1/2 On 3/2 ) if c n θ π/2, where N = n + η with η := α + β + 1)/2, J α is the Bessel function, and c > is an arbitrary but fixed constant see [17, Theorem ,. 195]). Using that 2/π sin θ/θ 1 and cos θ/2 1 on [, π/2], and also Γn + α + 1)/n! n α, we infer from above sin θ 2 ) 2α [ P α,β) k cos θ)] 2 c1 [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 ) 1/2 2 [ J α z) = cosz + γ) + Oz ) ], z, πz

27 26 GEORGE KYRIAZIS, PENCHO PETRUSHEV, AND YUAN XU where γ = απ/2 π/4. All of the above leads to 9.5) sin θ 2) 2α Λ n cos θ) n+[εn] k=n c nθ ) c 1 [J α k + η)θ)] 2 c 2 k 3/2 θ 1/2 J α k + η)θ) n+[εn] k=n [ cos 2 kθ + bθ)) c nθ) ] c εn, for c n θ π/2, where bθ) = 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 n + 1)θ cosn + m)θ m n + 1) + cos 2b + sin 2b)sinm. 2 2 sin θ k=n Therefore, n+[εn] k=n cos 2 kθ + bθ)) 1 2 [εn] + 1) 1 2 ) 1 [εn] + 1) sin θ 2 εn 1 π ) εn εnθ 4, whenever 2π/ε)n 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 >, if c n θ π/2 with c := max{c, 8c, 2π/ε} and n is sufficiently large. Hence, Λ n cos θ) c sin θ 2α 9.7) cθ θ 2) 2α+1), c n θ π/2, for sufficiently large n, which yields 2.6) in this case. For the remaining case δ 2 n θ c n, we need further roerties of Jacobi olynomials. Let x ν,n = cos θ ν,n denote the zeros of Jacobi olynomial P n α,β), where < θ 1,n < θ 2,n < < θ n,n < π. It is well known that θ ν,n ν/n, but we will need much more recise asymtotic reresentation for θ ν,n, see below. The Jacobi olynomials satisfy the following relation see e.g. [11, Theorem 3.3,. 171]), 9.8) P α,β) n cos θ) n 1/2 θ θ νθ,n nα+1/2, θ [, π], ν α+1/2 θ where ν θ denotes the index of the zero x ν,n, 1 ν n, which is one of) the closest to x x = cos θ). We will need the asymtotics of the zeros of the Jacobi olynomials from [5]: 9.9) θ ν,n = j α,ν N + 1 4N 2 [ α ) 1 t cot t 2t α2 β 2 4 tan t 2 ] + t 2 On 3 ), where N = n+η as before, j α,ν is the νth ositive zero of the Bessel function J α x) and t = j α,ν /N. Here the O-term is uniformly bounded for ν = 1, 2,..., [γn], where

28 JACOBI DECOMPOSITION OF WEIGHTED TRIEBEL-LIZORKIN AND BESOV SPACES 27 γ, 1). It is easy to verify that 1 t cot t)/t = Ot) as t and obviously 1/n + η) 1/n = On 2 ). Hence 9.1) θ ν,n = j α,ν n + On 2 ), ν = 1, 2,..., [γn]. We will also use that < j α,1 < j α,2 < and j α,ν. Let j α,νmax := max{j α,ν : j α,ν 1+ε)c } and denote J := {j α,1, j α,2,..., j α,νmax }. Notice that ν max is a constant indeendent of n. Suose that J = the case J = is easier). Fix δ 2 n θ c n. Then by 9.8) it follows that P α,β) k cos θ) k α+1 θ θ νθ,k, where the ν θ s involved are bounded by a constant indeendent of n. Hence, 9.1) can be used to reresent θ νθ,k for n k n + [εn] if n is sufficiently large. Using the above we get n+[εn] Λ n cos θ) cn 2α+2 k=n n+[εn] cn 2α cn k=n n+[εn] 2α k=n where we used 9.1). Therefore, 9.11) Λ n cos θ) cn n+[εn] θ θ νθ,k 2 cn 2α k=n kθ kθ νθ,k 2 ) kθ j α,νθ 2 c k kθ j α,νθ n+[εn] 2α ) kθ j α,νθ 2 c c ε, k=n dist kθ, J ) 2 c ), c, c >, where dist kθ, J ) denotes the distance of kθ from the set J, that is the distance of kθ from the nearest zero of the Bessel function J α x). It remains to estimate the sum in 9.11). Denote K := {n, n + 1,..., n + [εn]} and let K be the set of all indices k K such that dist kθ, J ) < mθ, where m := [εn/6ν max )]. Evidently #K 2m + 1)ν max 2[εn/6ν max )] + 1)ν max εn/2 if n 6ν max ε. Then #K \ K [εn] + 1 εn/2 εn/2 and hence n+[εn] k=n dist kθ, J ) 2 k K\K mθ) 2 c Inserting this in 9.11) we obtain Λ n cos θ) cn 2α c n c ) cn 2α+1 k K\K nθ) 2 cδ 2 2εn c n, c >. for sufficiently large n. This imlies the stated inequality 2.6) with x = cos θ in the case δ 2 n θ c n. The roof of Proosition 2.4 is comlete.