DECOMPOSITION OF SPACES OF DISTRIBUTIONS INDUCED BY HERMITE EXPANSIONS

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1 DECOMPOSITION OF SPACES OF DISTRIBUTIONS INDUCED BY HERMITE EXPANSIONS PENCHO PETRUSHEV AND YUAN XU Abstract. Decomoston systems wth radly decayng elements needlets) based on Hermte functons are ntroduced and exlored. It s roved that the Trebel-Lzorkn and Besov saces on R d nduced by Hermte exansons can be characterzed n terms of the needlet coeffcents. It s also shown that the Hermte Trebel-Lzorkn and Besov saces are, n general, dfferent from the resectve classcal saces. 1. Introducton The urose of ths aer s to extend the fundamental results of Frazer and Jawerth [4, 5] on the ϕ-transform to the case of Hermte exansons on R d. In the srt of [4, 5] we wll construct a ar of dual frames n terms of Hermte functons and use them to characterze the Hermte-Trebel-Lzorkn and Hermte- Besov saces. Let {h n } n=0 be the L 2 R d ) normalzed unvarate Hermte functons see 2.1). The d-dmensonal Hermte functons are defned by H α x) := h α1 x 1 ) h αd x d ). Then the kernel of the orthogonal roector of L 2 onto W n := san {H α : α = n} s gven by H n x, y) := α =n H αx)h α y). Our constructon of Hermte frames hnges on the fundamental fact that for comactly suorted C functons â the kernels Λ n x, y) := â n )H x, y) decay radly away from the man dagonal n R d. Ths fact was establshed n [3] for dmenson d = 1 and n [1] n general. We obtan a more recse estmate n Theorem 2.2 below. We utlze kernels of such knd for the constructon of a ar of dual frames {ϕ ξ } ξ X, {ψ ξ } ξ X, where X s a multlevel ndex set. The frame elements have almost exonental localzaton see 3.11)) whch romted us to call them needlets. The needlet systems of ths artcle can be vewed as an analogue of the ϕ-transform of Frazer and Jawerth [4, 5]. Frames of the same nature n the case d = 1 have been revously ntroduced n [3]. Our rmary goal s to utlze needlets to the characterzaton of the Trebel- Lzorkn and Besov saces n the context of Hermte exansons. To be more secfc, assume that â C, su â [1/4, 4], and â > c on [1/3, 3], and defne 1.1) Φ 0 := H 0 and Φ := ν=0 ν ) â 4 1 H ν, Mathematcs Subect Classfcaton. 42B35, 42C15. Key words and hrases. Localzed kernels, frames, Hermte olynomals, Trebel-Lzorkn saces, Besov saces. The second author was artally suorted by the NSF under Grant DMS

2 2 PENCHO PETRUSHEV AND YUAN XU Then for all arorate ndces we defne the Hermte-Trebel-Lzorkn sace F H) as the set of all temered dstrbutons f such that F f F := 2 α Φ f ) ) q) 1/q <, = where Φ fx) := f, Φx, ) see Defnton 4.1). We defne the Hermte-Besov saces B = B H) by the norm ) q ) 1/q. f B := 2 α Φ f One normally uses bnary dlatons n 1.1) see e.g. [17, 10.3] and also [1, 2]). We dlate â by factors of 4 nstead snce then the Hermte F- and B-saces embed ust as the classcal F- and B-saces. Our man results assert that the Hermte-Trebel-Lzorkn and Hermte Besov saces can be characterzed n terms of resectve sequence norms of the needlet coeffcents of the dstrbutons Theorems 4.5, 5.7). Furthermore, we use these results to show that the Hermte-F- and B-saces of essentally ostve smoothness are dfferent from the resectve classcal F- and B-saces on R d. Our develoment here s a art of a bgger roect for needlet characterzaton of Trebel-Lzorkn and Besov saces on nonclasscal domans such as the unt shere [10], the nterval wth Jacob weghts [7], and the unt ball [8]. The rest of the aer s organzed as follows: Secton 2 contans some background materal. The needlets are ntroduced n 3. In 4 the Hermte-Trebel-Lzorln saces are defned and characterzed va needlets. The Hermte-Besov saces are ntroduced and characterzed n 5. Secton 6 contans the roofs of a number of lemmas and theorems from 2-5. Some useful notaton: f := f L R ); for a measurable set E R d, E d denotes the Lebesgue measure of E and 1 E s the characterstc functon of E. Also, for x R d, x s the Eucldean norm of x, x := max 1 d x, and dx, E) := nf y E x y s the l dstance of x from E R d. Postve constants are denoted by c, c 1,... and they may vary at every occurrence; A B means c 1 A B c 2 A. 2. Prelmnares 2.1. Localzed kernels nduced by Hermte functons. We begn wth a revew of some basc roertes of Hermte olynomals and functons. For background nformaton we refer the reader to [16].) The Hermte olynomals are defned by H n t) = 1) n e t2 d ) n e t2), n = 0, 1,.... dt These olynomals are orthogonal wth resect to e t2 on R. We wll denote the L 2 -normalzed Hermte functons by One has R h n t) := 2 n n! π ) 1/2 Hn t)e t2 /2. h n t)h m t)dt = 2 n n! π ) 1 R H n t)h m t)e t2 dt = δ n,m.

3 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 3 As s well known the Hermte functons form an orthonormal bass for L 2 R). As already mentoned, the d-dmensonal Hermte functons H α are defned by 2.1) H α x) := h α1 x 1 ) h αd x d ), α = α 1,..., α d ). Evdently e x 2 /2 H α x) s a olynomal of degree α := α α d. The Hermte functons form an orthonormal bass for L 2 R d ). Moreover, H α are egenfunctons of the Hermte oerator D := + x 2 and 2.2) DH α = 2 α + d)h α, where s the Lalacan. The oerator D can be wrtten n the form 2.3) D = 1 2 d A A + A A ), where A = + x, A = + x. x x =1 Let e denote the th coordnate vector n R d. Then the oerators A and A satsfy 2.4) A H α = 2α + 2) 1 2 Hα+e and A H α = 2α ) 1 2 Hα e. Combnng these two relatons shows that {H α } satsfy the recurrence relaton 2.5) x H α x) = and also 2.6) x H α x) = ) 1 α +1 2 ) 1 α H α+e x) + α 2 2 H α+e x) + α 2 ) 1 2 H α e x) ) 1 2 H α e x). Let W n := san {H α : α = n} and V n := n W. The kernels of orthogonal roectors on W n and V n are gven by 2.7) H n x, y) := n H α x)h α y) and K n x, y) := H x, y), α =n resectvely. An mortant role wll be layed by oerators whose kernels are obtaned by smoothng out the coeffcents of the kernel K n by samlng a comactly suorted C functon â. For our uroses we wll be consderng smoothng functons â that satsfy: Defnton 2.1. A functon â C [0, ) s sad to be admssble of tye a) f su â [0, 1 + v] v > 0) and ât) = 1 on [0, 1], and of tye b) f su â [u, 1 + v], where 0 < u < 1, v > 0. For an admssble functon â we consder the kernel 2.8) Λ n x, y) := â H x, y). n) It wll be crtcal for our further develoment that the kernels Λ n x, y) and ther dervatves decay radly away from the man dagonal y = x n R d R d :

4 4 PENCHO PETRUSHEV AND YUAN XU Theorem 2.2. Suose â s admssble n the sense of Defnton 2.1 and let α N d 0. Then for any k 1 there exsts a constant c k deendng only on k, α, d, and â such that 2.9) α x α Λ n α 2 [K n+[vn]+ α +k x, x)] 1 2 [K n+[vn]+k y, y)] 1 2 nx, y) c k. 1 + n 1 2 x y ) k Here the deendence of c k on â s of the form c k = ck, α, u, d) max 0 l k â l). We relegate the somewhat lengthy roof of ths theorem to 6.1. The functon ) λ n x) := K n x, x) s termed Chrstoffel functon and t s known see e.g. [9]) to have the followng asymtotc n dmenson d = 1: { 2.11) λ n x) n max 1/2 n 2/3, 1 x }) 1/2 2n unformly for n 1 and x 2n1 + c n 2/3 ), where c > 0 s any fxed constant. Consequently, for d = 1 we have { 2.12) K n x, x) n max 1/2 n 2/3, 1 x }) 1/2, x 2n1 + c n 2/3 ). 2n For d 2 one has see [16,. 70]) 2.13) H n x, x) cn d/2 1, x R d. Ths along wth 2.7) leads to 2.14) K n x, x) cn d/2, x R d, d 1. On the other hand, t s well known that see e.g. [16,. 26]) 2.15) h n x) ce γx2, x 4n + 2) 1/2, γ > 0, and h n cn 1/12, whch readly mly 2.16) K n x, x) ce γ x 2, f x := max 1 d x 4n + 2) 1/2, where γ > 0 deends only on d. Now, combnng 2.9) wth 2.14) and 2.16) settng γ := γ /2) we arrve at Corollary 2.3. Under the hyothess of Theorem 2.2 we have 2.17) α x α Λ n α +d 2 nx, y) c k 1 + n 1 2 x y ), x k Rd, 2.18) and 2.19) α x α Λ e γ x 2 nx, y) c k 1 + n 1 2 x y ), f x k 4n+[vn]+ α +k)+2) 1/2, α x α Λ e γ y 2 nx, y) c k 1 + n 1 2 x y ), f y k 4n + [vn] + k) + 2) 1/2.

5 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 5 Note that an estmate smlar to 2.17) s roved n [3] when d = 1 and α = 0 and n the general case n [1]. Estmate 2.9) s new. We now turn to a lower bound estmate. Theorem 2.4. Let â be admssble n the sense of Defnton 2.1 and ât) > c > 0 on [1, 1 + τ], τ > 0. Then for any ε > 0, R d Λ n x, y) 2 dy c n d/2 for x 1 ε) 21 + τ)n, where c > 0 deends only on τ, ε, c, and d. Ths theorem rovdes a lower bound for the range where estmate 2.17) wth α = 0) s shar. To ndcate the deendence of K n on d, we wrte K n,d = K n. Theorem 2.4 s an mmedate consequence of 2.12) and the followng lemma. Lemma 2.5. If 0 < λ < 1, 0 < ρ < 1, and d 1, then there exsts a constant c > 0 such that for n 2/λ n 2.20) Hmx, 2 x) cn d 1 2 K[ρn],1 t, t) f t := x 2 2n + 1. m=[1 λ)n] The roof of ths lemma s gven n Norm relaton. For future use we gve here the well known relaton between dfferent norms of functons from V n see e.g. [9]): For 0 <, q 2.21) g cn d 2 1/q 1/ g q for g V n, wth c > 0 deendng only on, q, and d. Ths estmate can be roved by means of the kernels from 2.8) wth â admssble of tye a) Cubature formula. In order to defne our frame elements, we need a cubature formula exact for roducts fg wth f, g V n. Such a formula, however, s readly avalable usng the Gaussan quadrature formula. Prooston 2.6. [15] Denote by t ν,n, ν = 1, 2,..., n, the zeros of the Hermte olynomal H n t). The Gaussan quadrature formula n 2.22) ft)e t2 dt w ν,n ft ν,n ), w ν,n := λ n t ν,n )e t2 ν,n, R ν=1 s exact for all olynomals of degree 2n 1. Here λ n ) s the Chrstoffel functon defned n 2.10). The roduct nature of e x 2 enables us to obtan the desred cubature formula on R d rght away. Prooston 2.7. Let ξ α,n := t α1,n,..., t αd,n) and λ α,n := d ν=1 λ nt αν,n). The cubature formula n n 2.23) fx)gx)dx λ α,n fξ α,n )gξ α,n ) R d α 1=1 α d =1 s exact for all f V l, g V m wth l + m 2n 1.

6 6 PENCHO PETRUSHEV AND YUAN XU We next record some well known roertes of the zeros of Hermte olynomals. Suose {ξ ν } are the zeros of H n t) wth n even) ordered so that 2.24) ξ n 2 < < ξ 1 < 0 < ξ 1 < < ξ n 2, ξ ν = ξ ν. From [9] we have ξ n 2n + 1 n 1/6 and unformly for ν n/2 1 2 { 2.25) ξ ν+1 ξ ν 1 n max 1/2 n 2/3, 1 ξ }) 1/2 ν. 2n Consequently, on account of 2.11) 2.26) λ n ξ ν ) ξ ν 1 ξ ν+1, ν n/2 1 ξ 0 := 0). By [15, )] 2.27) πν 1 2 ) 2n + 1) < ξ 1/2 ν < 4ν + 3, ν = 1,..., n/2. 2n + 1) 1/2 From ths and 2.25) we have, for any ε > 0, 2.28) ξ ν+1 ξ ν 1 n 1/2 f ν 1/2 ε)n, and 2.29) c 1 n 1/2 ξ ν ξ ν 1 c 2 n 1/6 f 1/2 ε)n < ν n/2. Here the constants deend on ε. It also follows by 2.25) that 2.30) ξ ν+1 ξ ν 1 ξ ν ξ ν 2, n/2 + 2 ν n/2 1. For the constructon of our frames n Secton 3 we need the cubature formulae from Prooston 2.7 wth 2.31) n = 2N, where N := [1 + 11δ)4/π) 2 4 ] + 3 and 0 < δ < 1/37 s an arbtrary but fxed) constant. Gven 0, let as above ξ ν, ν = ±1,..., ±N, be the zeros of H 2N t). Let X be the set of all nodes of cubature 2.23) wth n = 2N,.e. X s the set of all onts ξ α := ξ α1,..., ξ αd ), where 0 < α ν N. Also, for ξ = ξ α we denote brefly λ ξ := λ α,n. Note that #X = 2N ) d 4 d. An mmedate consequence of Prooston 2.7 s the followng Corollary 2.8. The cubature formula 2.32) fx)gx)dx λ ξ fξ)gξ), λ ξ := R d ξ X s exact for all f V l, g V m wth l + m 4N 1. d λ 2N ξ αν ), For later use we now ntroduce tles {R ξ } nduced by the onts of X. Set I 1 := [0, ξ 1 + ξ 2 )/2], I 1 := I 1, ν=1 I ν := [ξ ν 1 + ξ ν )/2, ξ ν + ξ ν+1 )/2], ν = ±2,..., ±N 1, and I N := [ξ N 1 + ξ N )/2, ξ N + 2 /6 ], I N := I N. For each ξ = ξ α = ξ α1,..., ξ αd ) n X we set 2.33) R ξ := I α1 I α2 I αd,

7 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 7 and also 2.34) Q := [ξ N 2 /6, ξ N + 2 /6 ] d = ξ X R ξ. Thus we have assocated to each ξ X 0) a tle R ξ so that dfferent tles do not overla have dsont nterors) and they cover the cube Q [ 2, 2 ] d. Observe that by the constructon of the tles {R ξ } and 2.26) we have 2.35) λ ξ R ξ, ξ X. By 2.28) R ξ 2 d f ξ = ξ α X wth α 1/2 δ/2)2n = 1 δ)n. Assume that ξ α 1 + 4δ)2 +1. By 2.27) ξ α > π α 1/2) 4N + 1) 1/2 and hence π α 1/2) 4N + 1) 1/2 < 1 + 4δ)2+1. Usng the defnton of N n 2.31) t s easy to show that the above nequalty mles α 1 δ)n. Consequently, for ξ X, 2.36) R ξ ξ + [ 2, 2 ] d and R ξ 2 d f ξ 1 + 4δ)2 +1. On the other hand, by 2.28)-2.29) t follows that, n general, 2.37) ξ + [ c 1 2, c 1 2 ] d R ξ ξ + [ c 2 2 /3, c 2 2 /3 ] d, ξ X, and hence 2.38) c 2 d R ξ c 2 d/3. Fnally, note that snce the zeros of H n and H n+1 nterlace, each R η X +l, l 1, may ntersect at most fntely many deendng only on d) tles R ξ, ξ X Maxmal oerator. Let M s be the maxmal oerator, defned by 1/s ) M s fx) := su fy) dy) s, x R d, Q: x Q Q Q where the su s over all cubes Q n R d wth sdes arallel to the coordnate axes whch contan x. We wll need the Fefferman-Sten vector-valued maxmal nequalty see [14]): If 0 < <, 0 < q, and 0 < s < mn{, q}, then for any sequence of functons f 1, f 2,... on R d 2.40) ) 1/q [M s f )] q c f ) q) 1/q, =1 where c = c, q, s, d) Dstrbutons on R d. As s customary, we wll denote by S the Schwartz class of all functons φ C R d ) such that 2.41) P β,γ φ) := su x γ D β φx) < for all γ, β. x The toology on S s defned by the sem-norms P β,γ. Then the sace S of all temerate dstrbutons s defned as the set of all contnuous lnear functonals on S. The arng of f S and φ S wll be denoted by f, φ := fφ) whch s consstent wth the nner roduct f, g := R d fgdx n L 2 R d ). As a convenent notaton we ntroduce the followng convoluton : =1

8 8 PENCHO PETRUSHEV AND YUAN XU Defnton 2.9. For functons Φ : R d R d C and f : R d C, we wrte 2.42) Φ fx) := Φx, y)fy) dy. R d More generally, assumng that f S and Φ : R d R d C s such that Φx, y) belongs to S as a functon of y Φx, ) S), we defne Φ f by 2.43) Φ fx) := f, Φx, ), where on the rght f acts on Φx, y) as a functon of y. We next record some roertes of the above convoluton that are well known and easy to rove. Lemma a) If f S and Φ, ) SR d R d ), then Φ f S. Furthermore H n f V n. b) If f S, Φ, ) SR d R d ), and φ S, then Φ f, φ = f, Φ φ. c) If f S, Φ, ), Ψ, ) SR d R d ), and Φy, x) = Φx, y), Ψy, x) = Ψx, y), then 2.44) Ψ Φ fx) = Ψx, ), Φ, ) f. Evdently the Hermte functons {H α } belong to the sace of test functons S. More mortantly the functons n S can be characterzed by the coeffcents n ther Hermte exansons. Denote 2.45) Pr φ) := n+1) r H n φ 2 = n+1) r φ, H α 2) 1/2, r 0. n=0 Lemma We have n=0 α =n 2.46) φ S φ, H α c k α + 1) k for all α and all k. Moreover, the toology n S can be equvalently defned by the sem-norms Pr from above. Proof. a) Assume frst that the rght-hand sde estmates n 2.46) hold. Alyng reeatedly denttes 2.5)-2.6) one easly derves the estmate 2.47) su x γ D β H α x) c α + 1) γ + β )/2 max x ω α + β + γ H ω for all ndces β and γ, whch mles φ S usng that φ = n=0 n L 2. α =n φ, H α H α b) Suose φ S. Usng 2.2) we have 1 φ, H α = H α x)φx)dx = + x 2 )H α x)φx)dx R 2 α + d d R d 1 = H α x) φ + x 2 φx))dx, 2 α + d R d where for the last equalty we used ntegraton by arts. Reeatng the above rocedure k tmes we obtan a reresentaton for φ, H α of the form ) φ, H α = 2 α + d) k H α x) C β,γ x γ D β φx)dx R d β 2k, γ 2k whch yelds the rght-hand sde estmates n 2.46).

9 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 9 The equvalence of the toologes n S nduced by the sem-norms from 2.41) and 2.45) follows easly by 2.47) and 2.48). 3. Constructon of buldng blocks Needlets) We utlze the localzed kernels from Theorem 2.2 and the cubature formula from Corollary 2.8 to the constructon of a ar of dual frames consstng of localzed functons on R d. Let â, b satsfy the condtons: 3.1) â, b C R), su â, su b [1/4, 4], 3.2) ât), bt) > c > 0 f t [1/3, 3], 3.3) ât) bt) + â4t) b4t) = 1 f t [1/4, 1]. Consequently, 3.4) â4 ν t) b4 ν t) = 1, t [1, ). ν=0 It s easy to see that see e.g. [5]) f â satsfes 3.1)-3.2), then there exsts b satsfyng 3.1)-3.2) such that 3.3) holds true. Assumng that â, b satsfy 3.1)-3.3), we defne ν ) 3.5) Φ 0 := H 0, Φ := â 4 1 H ν, 1, and 3.6) Ψ 0 := H 0, Ψ := ν=0 ν=0 ν ) b 4 1 H ν, 1. Let X be the set of the nodes of cubature formula 2.32) from Corollary 2.8 and let λ ξ be the coeffcents of that cubature formula. We now defne the th level needlets by 3.7) ϕ ξ x) := λ 1/2 ξ Φ x, ξ) and ψ ξ x) := λ 1/2 ξ Ψ x, ξ), ξ X. Wrte X := X, where equal onts from dfferent levels X are consdered as dstnct elements of X. We use X as an ndex set to defne a ar of dual needlet systems Φ and Ψ by 3.8) Φ := {ϕ ξ } ξ X, Ψ := {ψ ξ } ξ X. Accordng to ther further roles, we wll call {ϕ ξ } analyss needlets and {ψ ξ } synthess needlets. The almost exonental localzaton of the needlets wll be crtcal for our further develoment. Indeed, by 2.17) we have 3.9) Φ ξ, x), Ψ ξ, x) c k 2 d x ξ ) k, x Rd, k, Fx L > 0. Then by 2.19) t follows that for any k > ) Φ ξ, x), Ψ ξ, x) Here c k deends on L and δ as well. c k 2 L x ξ ) k, x Rd, f ξ > 1 + δ)2 +1.

10 10 PENCHO PETRUSHEV AND YUAN XU From above and 2.35)-2.37) we nfer 3.11) ϕ ξ x), ψ ξ x) and 3.12) ϕ ξ x), ψ ξ x) c k 2 d/ x ξ ) k, f ξ 1 + δ)2 +1, c k 2 L x ξ ) k, f ξ > 1 + δ)2 +1. The followng rooston rovdes a dscrete decomoston of S and L R d ) va needlets. Prooston 3.1. a) If f S, then 3.13) f = Ψ Φ f n S and 3.14) f = ξ X f, ϕ ξ ψ ξ n S. b) If f L, 1, then 3.13) 3.14) hold n L. 1 < <, then the convergence n 3.13) 3.14) s uncondtonal. Moreover, f Proof. a) By the defnton of Φ and Ψ n 3.5)-3.6) t follows that Ψ 0 Φ 0 = H 0 and 4 ν ν ) Ψ Φ x, y) = ) b H ν x, y), 1. ν=4 2 â Note that Ψ x, y) and Φ x, y) are symmetrc functons e.g. Ψ y, x) = Ψ x, y)) snce H ν x, y) are symmetrc and hence Ψ Φ x, y) s well defned. Now, 3.4) and Lemma 2.11 yeld 3.13). To establsh 3.14), we note that Ψ x, ) and Φ y, ) belong to V 4 and alyng the cubature formula from Corollary 2.8, we obtan Ψ Φ x, y) = Ψ x, u)φ y, u) dy R d = λ ξ Ψ x, ξ)φ y, ξ) = ψ ξ x)ϕ ξ y). ξ X ξ X Consequently, Ψ Φ f = ξ X f, ϕ ξ ψ ξ. Ths along wth 3.13) mles 3.14). b) Reresentaton 3.13) n L follows easly by the rad decay of the kernels of the nth artal sums. We omt the detals. Then 3.14) n L follows as above. The uncondtonal convergence n L, 1 < <, follows by Prooston 4.3 and Theorem 4.5 below. Remark 3.2. It s well known that there exsts a functon â 0 satsfyng 3.1) 3.2) such that â 2 t)+â 2 4t) = 1, t [1/4, 1]. Suose that n the above constructon

11 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 11 b = â and â 0. Then ϕξ = ψ ξ. Now 3.14) becomes f = ξ X f, ψ ξ ψ ξ. It s easy to see that ths reresentaton holds n L 2 and f L 2 = f, ψ ξ 2) 1/2, f L 2, ξ X.e. {ψ ξ } ξ X s a tght frame for L 2 R d ). 4. Hermte-Trebel-Lzorkn saces F-saces) In ths secton we ntroduce the analogue of Trebel-Lzorkn saces n the context of Hermte exansons followng the general aroach descrbed n [17, 10.3] and show that they can be characterzed va needlets. In our treatment of Hermte- Trebel-Lzorkn saces we wll utlze the scheme of Frazer and Jawerth from [5] see also [6]) Defnton of Hermte-Trebel-Lzorkn saces. Let the kernels {Φ } be defned by ν ) 4.1) Φ 0 := H 0 and Φ := â 4 1 H ν, 1, ν=0 where {H ν } are from 2.7) and â obeys the condtons: 4.2) 4.3) â C [0, ), su â [1/4, 4], ât) > c > 0, f t [1/3, 3]. Defnton 4.1. The Hermte-Trebel-Lzorkn sace F := F H), where α R, 0 < <, 0 < q, s defned as the set of all f S such that 4.4) f F := 2 α Φ f ) ) q) 1/q <, where the l q -norm s relaced by the su norm when q =. As wll be shown n Theorem 4.5, the above defnton of Trebel-Lzorkn saces s ndeendent of the secfc selecton of â satsfyng 4.2)-4.3) n the defnton of Φ n 4.1). Prooston 4.2. The Hermte-Trebel-Lzorkn sace F whch s contnuously embedded n S F S ). s a quas-banach sace Proof. We wll only establsh that F S. Then the comleteness of F follows by a standard argument usng n addton Fatou s lemma and Prooston 3.1. As n Defnton 4.1, let {Φ } be defned by a functon â obeyng 4.2)-4.3). As already ndcated there exsts a functon b such that 3.1)-3.3) hold. Let {Ψ } be defned as n 3.6) usng ths functon. After ths rearaton, let {ϕ ξ } and {ψ ξ } be needlet systems defned as n 3.7)-3.7) usng these {Φ } and {Ψ }. Let f F. By Prooston 3.1 f = Ψ Φ f n S and hence f, φ = Ψ Φ f, φ = Φ f, Ψ φ, φ S.

12 12 PENCHO PETRUSHEV AND YUAN XU Alyng the Cauchy-Schwarz nequalty and 2.21) we obtan, for 2, 4 Φ f, Ψ φ Φ f 2 Ψ φ 2 c2 d/ Φ f c2 f F P r φ), ν=4 2 H φ 2 whenever r α + d/ + 1. Ths leads to f, φ c f F P r φ), whch yelds the clamed embeddng. Prooston 4.3. We have the followng dentfcaton: 4.5) F 02 L, 1 < <, wth equvalent norms. The roof of ths rooston can be carred out as the roof of Prooston 4.3 n [10] n the case of shercal harmoncs and wll be omtted. It emloys the exstng L multlers for Hermte exansons see e.g. [16]) Needlet decomoston of Hermte-Trebel-Lzorkn saces. In the followng we wll use the multlevel set X := X from 3 and the tles {R ξ } ntroduced n 2.33). Defnton 4.4. Let α R, 0 < <, 0 < q. The Hermte-Trebel- Lzorkn sequence sace f s defned as the set of all sequences of comlex numbers s = {s ξ } ξ X such that 4.6) s f := 2 [ q ) 1/q s ξ R ξ 1/2 1 Rξ )] < ξ X wth the usual modfcaton when q =. Assumng that {ϕ ξ }, {ψ ξ } s a dual ar of analyss and synthess needlets see 3.7)-3.8)), we ntroduce the oerators: S ϕ : f { f, ϕ ξ } ξ X Analyss oerator) and T ψ : {s ξ } ξ X ξ X s ξψ ξ Synthess oerator). We now come to our man result on Hermte-Trebel-Lzorkn saces. Theorem 4.5. If α R and 0 < <, 0 < q, then the oerators S ϕ : F f and T ψ : f F are bounded and T ϕ S ϕ = Id. Consequently, assumng that f S, we have f F f and only f { f, ϕ ξ } f and 4.7) f F Furthermore, the defnton of F satsfyng 4.2) 4.3). { f, ϕ ξ } f. s ndeendent of the secfc selecton of â For the roof of ths theorem we adat some technques from [5]. Defnton 4.6. For any collecton of comlex numbers {a ξ } ξ X, we defne 4.8) a x) := a η η x ) σ η X and 4.9) a ξ := a ξ), ξ X, where σ > d s suffcently large and wll be secfed later on.

13 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 13 We wll need a coule of lemmas whose roofs are gven n 6. Lemma 4.7. Suose s > 0 and σ > d max{2, 1/s}. Let {b ω } ω X, 0, be a set of comlex numbers. Then ) 4.10) b x) cm s b ω 1 Rω x), x R d. ω X Moreover, for ξ X, 4.11) b ξ1 Rξ x) cm s ω X b ω 1 Rω Here the constants deend only on d, δ, σ, and s. Lemma 4.8. Let g V 4 and denote ) x), x R d. M ξ := su x R ξ gx), ξ X, and m λ := nf x R λ gx), λ X +l. Then there exsts l 1, deendng only d, δ, and σ, such that for any ξ X 4.12) M ξ cm λ for all λ X +l, R λ R ξ, and hence 4.13) M ξ 1 Rξ x) c λ X +l,r λ R ξ where c > 0 deends only on d, δ, and σ. m λ1 Rλ x), x R d, Proof of Theorem 4.5. Suose q < the case q = s easer) and ck s, σ, and k so that 0 < s < mn{, q} and k σ > d max{1, 1/s}. Let {Φ } be from the defnton of Hermte-Trebel-Lzorkn saces see 4.1)- 4.3)). As already ndcated n the begnnng of 3, there exsts a functon b satsfyng 3.1)-3.2) such that 3.3) holds as well. We use ths functon to defne {Ψ } exactly as n 3.6). We further use {Φ } and {Ψ } to defne ust as n 3.7) a ar of dual needlet systems {ϕ η } and {ψ η }. Let { ϕ η }, { ψ η } be a second ar of needlet systems, defned as n 3.5)-3.7) from another ar of kernels { Φ }, { Ψ }. Our frst ste s to establsh the boundedness of the oerator T ψ : f F, defned by T ψs := ξ X s ψ ξ ξ. Prooston 4.2 and the fact that fntely suorted sequences are dense n f mly that t suffces to rove the boundedness of T ψ only for fntely suorted sequence. So, assume s = {s ξ } ξ X s a fntely suorted sequence and let f := T ψs. Evdently Φ ψ ξ = 0 f ξ X ν and ν 2, and hence Φ f = +1 ν= 1 ξ X ν s ξ Φ ψ ξ X 1 := ). Let ξ X ν, 1 ν + 1, and ξ 1 + δ)2 ν+1. Then usng 3.9)-3.11) we get Φ ψ ξ x) c2 3d/2 1 R x y ) k ξ y ) k dy c2 d/ ξ x ) k. d

14 14 PENCHO PETRUSHEV AND YUAN XU Hence, on account of 2.36) 4.14) Φ ψ ξ x) c R ξ 1/ ξ x ) k, x Rd. If ξ X ν, 1 ν + 1, and ξ > 1 + δ)2 ν+1, then by 3.9)-3.12) Φ ψ ξ x) c2 L R d 2 d x y ) k ξ y ) k dy c2 L ξ x ) k for any L > 0. Consequently, n vew of 2.37), estmate 4.14) holds agan. Denote S ξ := s ξ R ξ 1/2. Then by 4.14) we have 4.15) Φ fx) c +1 ν= 1 +1 ν= 1 ξ X ν s ξ Φ ψ ξ x) c S νx) S 1 := 0), +1 ν= 1 ξ X ν s ξ R ξ 1/ ν ξ x ) k where S νx) s defned as n 4.8). We nsert ths n 4.4) and aly Lemma 4.7 and the maxmal nequalty 2.40) to obtan f F 2 α S ) ) q) 1/q [M s 2 )] q ) 1/q α s ξ R ξ 1/2 1 Rξ c {s η } f. ξ X Hence the oerator T ψ : f F s bounded. Assumng that the sace F s defned va {Φ } nstead of {Φ } we next rove the boundedness of the oerator S ϕ : F f. Let f F and set M ξ := su x R ξ Φ fx), ξ X, and m λ := nf x R λ Φ fx), λ X +l, where l s the constant from Lemma 4.8. We have f, ϕ ξ c R ξ 1/2 Φ fξ) c R ξ 1/2 M ξ c R ξ 1/2 M ξ. By Lemma 2.10, Φ f V 4, and alyng Lemma 4.8 see 4.13)), we have Mξ 1 Rξ x) c m λ1 Rλ x), x R d. λ X +l,r λ R ξ

15 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 15 We use the above, Lemma 4.7, and the maxmal nequalty 2.40) to obtan { f, ϕ ξ } f c 2 ) q ) 1/q M ξ 1 Rξ ξ X c 2 ) q ) 1/q m λ1 Rλ λ X +l ) q ) 1/q c M s 2 α m λ 1 Rλ λ X +l ) q ) 1/q c 2 α m λ 1 Rξ λ X +l c 2 Φ f q) 1/q = c f F. Here for the second nequalty we used that each tle R λ, λ X +l, ntersects no more that fntely many deendng only on d) tles R η, η X. The above confrms the boundedness of the oerator S ϕ : F f. The dentty T ψ S ϕ = Id follows by Theorem 3.1. We fnally show the ndeendence of the defnton of Trebel-Lzorkn saces from the secfc selecton of â satsfyng 4.2)-4.3). Let {Φ }, { Φ } be two sequences of kernels as n the defnton of Trebel-Lzorkn saces defned by two dfferent functons â satsfyng 4.2)-4.3). As n the begnnng of ths roof, there exst two assocated needlet systems {Φ }, {Ψ }, {ϕ ξ }, {ψ ξ } and { Φ }, { Ψ }, { ϕ ξ }, { ψ ξ }. Denote by f F Φ) and f F Φ) the F -norms defned va {Φ } and { Φ }. Then from above t follows that f F Φ) c { f, ϕ ξ } f c f. F Φ) The clamed ndeendence of the defnton of F of the secfc selecton of â n the defnton of the functons {Φ } follows by nterchangng the roles of {Φ } and { Φ } and ther comlex conugates. The Hermte-F-saces embed n one another smlarly as the classcal F-saces. Prooston 4.9. a) If 0 < <, 0 < q, q 1, α R and ε > 0, then 4.16) F α+ε,q F 1. b) Let 0 < < 1 <, 0 < q, q 1, and < α 1 < α <. Then we have the contnuous embeddng 4.17) F F α1q1 1 f α d/ = α 1 d/ 1. The roof of ths embeddng result uses estmate 2.21) and Theorem 4.5 and can be carred out exactly as n the classcal case on R n see e.g. [17],. 47 and. 129). We omt t Comarson of Hermte-F-saces wth classcal F-saces. We next use needlet decomostons to show that the Hemte-Trebel-Lzorkn saces of essentally ostve smoothness are dfferent from the corresondng classcal Trebel- Lzorkn saces on R d.

16 16 PENCHO PETRUSHEV AND YUAN XU Theorem Let 0 < <, 0 < q, and α > d1/ 1) +. Then there exsts a functon f F such that f F H) = and hence f F H). Here F and F H) are the resectve classcal and Hermte Trebel-Lzorkn saces. Proof. For any y R d and a functon f we defne 4.18) f F y := q ) 1/q 2 R ξ 1/2 f, ϕ ξ 1 Rξ )). ξ X, ξ y > y /2 Choose a functon h C R d ) such that h = 1 and su h B0, 1), where B0, 1) := {x R d : x < 1}. Theorem 4.10 wll follow easly by the followng lemma whose roof s gven n 6.2. Lemma Wth the notaton from above, we have 4.19) 4.20) h y) F H) as y, and h y) F y 0 as y. By ths lemma t follows that there exsts a sequence {y } 1 R d such that 0 < y 1 < y 2 <... and y +1 > 3 y, h y ) F α q H) > 2 2, and h y ) F y < 1, = 1, 2,.... We now defne fx) := =1 f x), where f x) := 2 hx y ), x R d. Set τ := mn{, q, 1}. Evdently, h belongs to all classcal Trebel-Lzorkn saces, whch are shft nvarant, and hence f τ Fq α 2 τ h y ) τ Fq α = h τ Fq α 2 τ c h τ Fq α <. =1 Here we use that g τ Fq α On the other hand, for any l 1, f τ F α q g τ F α q H) c 2 ν ν=0 ξ X ν, ξ y l y l /2 c f l τ Fq α H) ) f τ Fy =1 = c2 lτ h y l ) τ F α q > c2 lτ c =1. Thus f F α q. ) q ) 1/q R ξ 1/2 f, ϕ ξ 1 τ Rξ ) H) c 2 τ h y ) τ Fy =1 2 τ c2 lτ c. =1 Here for the second nequalty we used that f ξ y l y l /2, then ξ y > y /2 for all l. Consequently, f τ F α q H) =. 5. Hermte-Besov saces B-saces) Besov tye saces are natural to ntroduce n the context of Hermte exansons see e.g. [17, 10.3]). We wll call them Hermte-Besov saces. To characterze these sace va needlets we use the aroach of Frazer and Jawerth [4] see also

17 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 17 [6]) to the classcal Besov saces. We refer to [11, 17] as general references for Besov saces Defnton of Hermte-Besov saces. Defnton 5.1. Let the kernels {Φ } be defned by 4.1) wth â satsfyng 4.2) 4.3). The Hermte-Besov sace B := B H), where α R, 0 <, q, s defned as the set of all f S such that ) q ) 1/q 5.1) f B := 2 α Φ f <, where the l q -norm s relaced by the su-norm f q =. Smlarly as for Hermte-Trebel-Lzorkn saces 4) Theorem 5.3 below mles that the above defnton of Hermte-Besov saces s ndeendent of the secfc selecton of â; also B s a quas-banach sace whch s contnuously embedded n S Needlet decomoston of Hermte-Besov saces. As for the Hermte- Trebel-Lzorkn saces we emloy the tles {R ξ } ntroduced n 2.33) n the followng. Also as before X := X. Defnton 5.2. The Hermte-Besov sequence sace b, where α R, 0 <, q, s defned as the set of all sequences of comlex numbers s = {s ξ } ξ X such that 5.2) s b := [2 α ) 1/ ] q ) 1/q R ξ 1 /2 s ξ < ξ X wth obvous modfcatons when = or q =. In the followng, we assume that {Φ }, {Ψ }, {ϕ ξ }, {ψ ξ } s a needlet system defned by 3.5)-3.8). Recall the analyss oerator: S ϕ : f { f, ϕ ξ } ξ X, and the synthess oerator: T ψ : {s ξ } ξ X ξ X s ξψ ξ. Theorem 5.3. If α R and 0 <, q, then the oerators S ϕ : B b and T ψ : b B are bounded and T ϕ S ϕ = Id. Consequently, assumng that f S, we have f B 5.3) f B Furthermore, the defnton of B 4.2) 4.3). f and only f { f, ϕ ξ } b { f, ϕ ξ } b. and s ndeendent of the choce of â satsfyng For the roof of ths theorem we need one addtonal lemma. Lemma 5.4. For any g V 4, 0, and 0 < 5.4) R ξ max gx) ) 1/ c g. x R ξ ξ X The roof of ths lemma s gven n 6. Proof of Theorem 5.3. Let 0 < s < and σ > d max{1, 1/s}. Just as n the roof of Theorem 4.5 we assume that {Φ }, {Ψ }, {ϕ η }, {ψ η } and { Φ }, { Ψ }, { ϕ η }, { ψ η } are two needlet systems, defned as n 3.5)-3.7), whch orgnate from two comletely dfferent functons â satsfyng 4.2)-4.3).

18 18 PENCHO PETRUSHEV AND YUAN XU We frst rove the boundedness of the oerator T ψ : b B, defned by T ψs := ξ X s ψ ξ ξ, assumng that B s defned by {Φ }. As n the Trebel- Lzorkn case due to the embeddng B S t suffces to consder only the case of a fntely suorted sequence s = {s ξ } ξ X. Let f := T ψs. By 4.15) and Lemma 4.7 we get Φ f c whch leads to f B c +1 ν= 1 +1 ν= 1 c {s η } b M s ω X ν R ω 1/2 s ω 1 Rω ) ) 1/ R ω 1 /2 s ω X 1 := ), ω X ν To rove the boundedness of the oerator S ϕ : B s defned n terms of {Φ }. Observng that and hence to the boundedness of T ψ. b f, ϕ ξ = λ 1/2 ξ Φ fξ) R ξ 1/2 Φ fξ), ξ X, we assume that B and Φ f V 4, we get usng Lemma 5.4 ) 1/ ) 1/ R ξ 1 /2 f, ϕ ξ c R ξ Φ fξ) c Φ f. ξ X ξ X Ths yelds { f, ϕ ξ } b c f B and hence the oerator S ϕ s bounded. The dentty T ψ S ϕ = Id s a consequence of Prooston 3.1. The ndeendence of the defnton of B from the artcular selecton of â follows from above exactly as n the case of Trebel-lzorkn saces see the roof of Theorem 4.5). The Hermte-Besov saces embed smlarly as the classcal Besov saces. Prooston 5.5. a) If 0 <, q, q 1, α R and ε > 0, then 5.5) B α+ε,q B 1. b) Let 0 < < 1 <, 0 < q, and < α 1 < α <. Then we have the contnuous embeddng 5.6) B B α1q 1 f α d/ = α 1 d/ 1. c) If 0 < <, 0 < q, α R, then 5.7) B α,mn{,q} F B α,max{,q}. Part b) of ths rooston follows readly by estmate 2.21). The roofs of arts a) and c) are as n the classcal case. We now show that under some restrcton on the ndces the Hemte-Besov saces are essentally dfferent from the classcal Besov saces on R d. Theorem 5.6. Let 0 <, q <, and α > d1/ 1) +. Then there exsts a functon f B such that f B H) = and then f B H). Here B and H) are the resectve classcal and Hermte Besov saces. B

19 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 19 Proof. We roceed qute smlarly as n the roof of Theorem Gven y R d and a functon f we defne 5.8) f B y := 2 R ξ 1 /2 f, ϕ ξ ) q/) 1/q. ξ X, ξ y > y /2 Pck h C R d ) such that h = 1 and su h B0, 1). The theorem follows easly by the followng: 5.9) 5.10) h y) B H) as y, and h y) B y 0 as y. To rove 5.9) we wll show that there exst ε > 0 and r > 1 such that 5.11) B H) Fr ε2 H). Then the result follows by the argument from the roof of Lemma Let > 1. Pck ε > 0 so that α > 2ε. Then by Proostons 4.9, 5.5 we have the followng embeddngs B B 2ε, F 2ε, F ε2 whch confrms 5.11). Let 1. Then α > d1 1/) and hence as n the roof of Lemma 4.11) there exst ε, δ > 0 such that α d/ = 3ε d/1 + δ). Then Proostons 4.9, 5.5 gve us the followng embeddngs B B 3ε,q 1+δ B2ε,1+δ 1+δ F 2ε,1+δ 1+δ F1+δ ε2, whch leads agan to 5.11). The roof of 5.10) s smlar to the roof of 4.20) and wll be omtted. We fnally want to lnk the Hermte-Besov saces wth the L -aroxmaton from lnear combnatons of Hermte functons. Denote by E n f) the best aroxmaton of f L from V n,.e. 5.12) E n f) := nf g V n f g. Let A be the aroxmaton sace of all functons f L for whch 2 α E 2 f) ) q) 1/q < 5.13) f A := f + wth the usual modfcaton when q =. Prooston 5.7. If α > 0, 1, 0 < q, then B equvalent norms. = A α/2,q Proof. Let f B. We frst observe that under the condtons on α,, and q, B s contnuously mbedded n L,.e. f can be dentfed as a functon n L and f c f B. The roof of ths s easy and standard and wll be omtted. By a well known and easy constructon there exsts a functon â 0 satsfyng 4.2)-4.3) such that ât)+â4t) = 1 for t [1/4, 1] and hence ν=0 â4 ν t) = 1 for t [1, ). Assume that {Φ } are defned by 4.1) usng ths functon â. By Theorem 5.3 the defnton of the Besov saces B s ndeendent of the selecton of â and hence they can be defned va these functons {Φ }. Smlarly as n Prooston 3.1 f = Φ f for f L. Now, usng that Φ f V 4, we have E 4 mf) =m+1 Φ f, m 0. A standard argument emloyng ths leads to the estmate f A c f A α/2,q B. To rove the estmate n the other drecton, let g V 4 2 2). Evdently, Φ f = Φ f g) and the rad decay of Φ yelds Φ f c f g. wth

20 20 PENCHO PETRUSHEV AND YUAN XU Consequently, Φ f ce 4 2f), 2, and Φ f c f. These lead to f B c f A. A α/2,q 6.1. Proofs for Secton Proofs Proof of Theorem 2.2. Ths roof hnges on an mortant lemma from [16]. Let ψ be a unvarate functon. The forward dfferences of ψ are defned by ψt) = ψt + 1) ψt) and k ψ = k 1 ψ), k 2. For a gven functon ψ we defne M ψ x, y) := ψν)h ν x, y), and then M k ψ = k ψν)h ν. ν=0 Lemma 6.1. [16,. 72] Let A x) and A y) x and y varables, resectvely. Then for any k 1, 6.1) 2 k x y ) k M ψ x, y) = k/2 l k ν=0 denote the oerator A aled to the c l,k A y) where c l,k are constants gven by ) k c l,k = 1) k l 4 k l 2k 2l 1)!!. 2l k ) 2l k A x) M ψx, y), l Proof. Ths lemma s roved n [16] excet that the constants c l,k are not determned exlctly there. For k = 1 one has [16, )] ) 6.2) 2x y )M ψ x, y) = A y) A x) M ψ x, y). The general result s obtaned by nducton usng the dentty [16, )] ) r ) r ) r 1 6.3) x y ) A y) A x) A y) A x) x y ) = 2r A y) A x). Assume that 6.1) holds for some k 0. Then usng 6.3) we get 2 k+1 x y ) k+1 M ψ = 2x y ) c l,k A y) A x) = k/2 l k 42l k) = c k,k A y) + c l,k [ A y) k+1)/2 l k where c l,k := 0 f l < k/2. relatons A y) A x) k/2 l k ) 2l k 2x y )M l ψ ) 2l k 1 A x) M ψ] l ) k+1 A x) M k+1 ψ [c l 1,k 42l k)c l,k ] A y) ) 2l k M l ψ ) 2l k 1 A x) M ψ, l Consequently, the coeffcents satsfy the recurrence c k+1,k+1 = c k,k, c l,k+1 = c l 1,k 42l k)c l,k, k + 1)/2 l k.

21 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 21 It follows from ths and 6.2) that c k,k = 1 for all k. Furthermore, the recurrence relaton above shows that c k,k = 4 k 1 ν=2 1 ν 2 + 2)c ν +1,ν, from whch one uses nducton and the fact that k 1 ν ν= ) = k +1) to derve the stated dentty for c l,k. The case α = 0,..., 0). Assume k 2. By Lemma 6.1, we have 2 k x y ) k Λ n x, y) = c l,k k/2 l k ν=0 ν l â A n) y) ) 2l k A x) Hν x, y), where l â ν n ) s the lth forward dfference aled wth resect to ν. Note frst that alyng the Cauchy-Schwarz nequalty, we have H α+β x)h α+γ y) 2 H α+β x) 2 H α+γ y) 2 6.4) α =n α =n α =n H n+ β x, x)h n+ γ y, y). We next consder the acton of A on H x, y). Usng reeatedly 2.4) we get ) mhβ m 6.5) = [2β + r) + 2] 1/2 H β+me. A x) Ths along wth 6.4) leads to r=0 A y) ) A x) ) m H ν x, y) 2 = α =ν r=0 1 m 1 [2α + r) + 2] 1/2 [2α + r) + 2] 1/2 r=0 H α+e y)h α+me x) 2 [2ν + + m)] +m H α+e y)h α+me x) α =ν [2ν + + m)] +m H ν+ y, y)h ν+m x, x). Hence the bnomal theorem and the Cauchy-Schwarz nequalty gve ) 2l k 2l k A y) A x) ) 2l k A y) Hν x, y) ) A x) ) 2l k H ν x, y) =0 2l k ) 2l k 2ν + 4l 2k) 2l k)/2 [H ν+ y, y)] 1 2 [H ν+2l k x, x)] 1 2 [ 2l k cν 2l k)/2 =0 =0 ] 1 [ 2 2l k H ν+ y, y) H ν+ x, x) A well known roerty of the dfference oerator gves ν ) 6.6) l â = n l â l) ξ) n l â l). n =0 ]

22 22 PENCHO PETRUSHEV AND YUAN XU By Defnton 2.1 t follow that l â ν n ) = 0 f 0 ν un l or ν n + vn, where 0 < u 1 and v > 0. Here u = 1 f â s of tye a).) Usng ths, the above estmates, and the Cauchy-Schwarz nequalty we nfer x y ) k Λ n x, y) c c l,k n l â l) n 2l k)/2 k/2 l k n+[vn] ν=[un] l [ 2l k =0 ] 1 [ 2 2l k H ν+ y, y) H ν+ x, x) =0 c k n k/2 [ K n+[vn]+k y, y) ] 1 2 [ K n+[vn]+k x, x) ] 1 2, where c k > 0 s of the form c k = ck, u, d) max 0 l k â l). Consequently, 6.7) Λ n x, y) c k [ Kn+[vn]+k y, y) ] 1 2 [ K n+[vn]+k x, x) ] 1 2 n x y ) k, x y. We also need estmate Λ n x, y) whenever x and y are close to one another. Alyng the Cauchy-Schwarz nequalty to the sum n 2.7) whch defnes H ν we get Λ n x, y) n+[vn] ν=0 ν ) Hν â x, x) 1/2 H ν y, y) 1/2 n c â [ Kn+[vn] y, y) ] 1 2 [ K n+[vn] x, x) ] 1 2, whch couled wth 6.7) yelds 2.9) n the case under consderaton. The case α > 0. We wll make use of the relaton = x A, where as usual := x. In the followng we agan denote by A x) the oerator A actng on the x varables, and A 0 s understood as the dentty oerator; by defnton A x) ) α := A x) 1 )α 1... A x) d )α d. We also dentfy the oerator of multlcaton by x wth x. In order to use Lemma 6.1, we wll need two commutng relatons. Lemma 6.2. Let k, r, s be nonnegatve ntegers. Then ) k r ) 6.8) x r A x) A y) r k! ) k = A x) A y) x r k )! and 6.9) x y ) k A x) where k!/k )! := 0 f k <. =0 ) s = s =0 ) s k! k )! Proof. To rove 6.8) we start from the dentty: ) k 6.10) x A x) A y) = k A x) A y) For k = 1 ths follows from the obvous denttes 6.11) x A x) In general t follows readly by nducton. A x) ) s x y ) k, ) k 1 + A x) = Id + A x) x and x A y) = A y) x. ) k A y) x. ] 1 2

23 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 23 We now roceed by nducton on r. Suose 6.8) holds for some r 1 and all k 1. Then ) k r ) x r+1 A x) A y) r k! ) k = x A x) A y) x r k )! =0 r ) [ r k! = A x) A y) k )! =0 r+1 [ ) )] r r k! = + A x) 1 k )! =0 whch comletes the nducton ste as ) r + r To roof 6.9), we start from x y ) k A x) ) k x r+1 + k ) A x) ) k A y) x r+1, 1 A y) ) ] k 1 x r ) = r+1 ). Thus 6.8) s establshed. = kx y ) k 1 + A x) x y ) k. For k = 1 ths dentty follows from 6.11) and, n general, by nducton. Fnally, one roves 6.9) by nducton on s smlarly as above. We omt the detals. The next lemma s nstrumental n the roof of Theorem 2.2 n the case α > 0. Lemma 6.3. If α, β N d 0 and k 1, then A x)) α [ x β n α + β 2 Kn+[vn]+ α + β +k x, x) ] 1 [ 2 K n+[vn]+k y, y) ] 1 2 Λ n x, y) c k 1 +. n x y ) k Proof. We frst show that for 1 d x y ) k A x)) α x β 6.12) Λ n x, y) c k n k+ α + β )/2 [ K n+[vn]+ α + β +k x, x) ] 1 [ 2 K n+[vn]+k y, y) ] 1 2. Clearly A x)) α x β = A x)) ) α α e x β β e A x) α x β and the two oerators searated by a dot commute. Usng 6.9) and Lemma 6.1, we have 6.13) 2 k x y ) k A x)) α x β Λ n x, y) = 2 k A x)) α α e x β βe α ) α k! ) α A x) x β k )! x y ) k Λ n x, y) α ) α 2 k! ν = c l,k â k )! n) k )/2 l k A x)) α e x β A y) ν=0 A x) ) 2l k+ Hν x, y). Furthermore, by 6.8), A x) ) α x β A y) ) 2l k+ β A x) = A x) µ=0 ) α A y) ) β 1) µ 2l k + )! µ 2l k + µ)! ) 2l k+ µ A x) x β µ.

24 24 PENCHO PETRUSHEV AND YUAN XU As A x)) α α e x β β e commutes wth A x) and x, we then conclude that A x)) α e x β A y) ) 2l k+ A x) Hν x, y) = A x)) α e A y) A x) β µ=0 ) β 1) µ 2l k + )! µ 2l k + µ)! ) 2l k+ µ x β µe H ν x, y). Usng relaton 2.5) reeatedly, t follows readly that r x r H λ x) = b m,r λ )H λ+r 2m)e x), m=0 where b m,r λ ) are ostve numbers satsfyng b m,r λ ) λ r/2. Alyng ths dentty to all varables we obtan 6.14) x β µe H ν x, y) = γ 1 ω 1=0... γ d ω d =0 b ω,γ λ) H λ+γ 2ω x)h λ y), λ =ν where γ = β for and γ = β µ, and b ω,γ λ) = b ω1,γ 1 λ 1 )... b ωd,γ d λ d ). Clearly, b ω,γ λ) cν β µ)/2. We now use the bnomal formula and 6.5) to obtan A x)) α e A y) c 2l k+ µ q=0 A y) A x) ) 2l k+ µ Hλ+γ 2ω x)h λ y) ) 2l k+ µ q A x)) α e +qe Hλ+γ 2ω x)h λ y) 2l k+ µ c ν α +2l k µ)/2 H λ+γ 2ω e+qe x)h λ+2l k+ µ q)e y) q=0 and hence A x)) α e x β A y) γ 1 ω 1 =0 γ d ω d =0 A x) 2l k+ µ q=0 λ =ν ) 2l k+ µ Hν x, y) β c µ=0 ν 2l k+ α + β 2µ)/2 H λ+γ 2ω+α e +qe x)h λ+2l k+ µ q)e y).

25 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 25 As before l â ν n ) = 0 f 0 ν un l or ν n + vn, where 0 < u 1 and v > 0. Also, by 6.6) l â ν n ) n l â ). We use all of the above to conclude that ) α x y ) k A x) x β Λ α nx, y) cn k+ α + β )/2 c l,k n+[vn] β γ 1 ν=[un] l µ=0 ω 1=0 [ 2l k+ µ q=0 γ d ω d =0 [ 2l k+ µ q=0 ] 1 2 H ν+2l k+ µ q y, y) k )/2 l k H ν+ α + γ 2ω +q x, x) cn k+ α + β )/2 [ K n+[vn]+ α + β +k x, x) ] 1 2 [ K n+[vn]+k y, y) ] 1 2, where we agan used the Cuachy-Schwarz nequalty. Ths roves 6.12). On the other hand, usng 6.14) wth µ = 0 and 6.5) we can wrte A x)) α x β Λ n x, y) = β 1 λ =ν ω 1=0... β d ω d =0 A x)) α n+[vn] ν=0 ν â n) b ω,β λ)c α λ) H λ+α+β 2ω x)h λ y), where c α λ) λ α /2. Hence, usng the fact that b ω,β λ) λ β /2 we conclude that A x)) α n+[vn] x β ν Λ n x, y) c â ν n) α + β )/2 ν=0 β 1 λ =ν ω 1 =0... β d ω d =0 H λ+α+β 2ω x)h λ y) cn α + β )/2 [ K n+[vn]+ α + β x, x) ] 1 2 [ K n+[vn] y, y) ] 1 2. Ths along wth 6.12) comletes the roof of Lemma 6.3. The last ste n the roof of Theorem 2.2 s to show that the oerator α can be reresented n the form 6.15) α = c βγ A β x γ, β+γ α where β+γ α means β +γ α for 1 d, and c βγ are constants deendng only on α, β, γ). By 2.3) = x A and hence r = x A ) r. The oerators x multlcaton by x ) and A do not commute, but t s easy to see that x s A = sx s 1 +A x s. Alyng ths reeatedly one fnds the reresentaton r = c νµ A ν x µ. 0 ν+µ r Snce the oerator A ν xµ commutes wth As xl f, ths readly mles reresentaton 6.15). ] 1 2

26 26 PENCHO PETRUSHEV AND YUAN XU Evdently, Lemma 6.3 and 6.15) yeld 2.9) whenever α > 0. The roof of Theorem 2.2 s comlete. Proof of Lemma 2.5. Observe frst that t suffces to rove 2.20) only for n suffcently large snce t holds trvally f 2/λ n c. We next rove 2.20) for d = 1. The Chrstoh-Darboux formula for the Hermte olynomals [15, 5.59)]) shows that K m x, x) = 2 m+1 m!) 1 [ H m+1x)h m x) H mx)h m+1 x) ]. Usng the fact that H m+1x) = 2m + 1)H m x) [15, )]) and H m+1 x) = 2xH m x) 2mH m 1 x) [15, 5.5.8)]), we can rewrte K m x, x) as K m x, x) = 2 m+1 m!) 1 [ 2m + 1)H 2 mx) 4mxH m 1 x)h m x) + 2m 2 H 2 m 1x) ]. Wrtten n terms of the orthonormal Hermte functons, the above dentty becomes m + 1)h 2 mx) + mh 2 m 1x) = K m x, x) + 2m x h m x)h m 1 x). In artcular, t follows that for x 2 2m + 1 m + 1)h 2 mx) + mh 2 m 1x) K m x, x) 2 2m 2m + 1 h m x) h m 1 x). and hence 3m + 2)h 2 mx) + 3mh 2 m 1x) K m x, x) + 2m + 1 hm x) 2m h m 1 x) Consequently, for x 2 2n + 1 n h 2 mx) 1 2 m=[1 λ)n] K m x, x). c n whch roves 2.20) when d = 1. n m=[1 λ)n] n m=[1 λ)n] h 2 m x) + h 2 m+1x) ) K m x, x) c 1 K [ρn] x, x), For d > 1 we need the followng dentty whch follows from the generatng functon of Hermte olynomals see e.g. [16]): H k x, x)r k = π d/2 1 r 2 ) d/2 e 1 r 1+r x 2 := F d r, t), k=0 where t = x. Let us denote H k,d x, x) = H k x, x) for x R d n order to ndcate the deendence on d. Then t follows from above that 6.16) r k [H k,d x, x) H k 2,d x, x)] = 1 r 2 ) r k H k,d x, x) k=0 k=0 = 1 r 2 )F d r, t) = π 1 F d 2 r, t). Notce that H n,d x, x) s a radal functon and hence a functon of t. Thus comarng the coeffcents of r k n both sde shows that H k,d x, x) H k 2,d x, x) = π 1 H k,d 2 x, x), ) 2

27 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 27 whch mles H k,d x, x) + H k 1,d x, x) = π 1 Now, summng over k we get n H k,d x, x) 1 2 k=[1 λ)n] c n k=[1 λ)n]+1 n k=[1 λ)n]+1 k H,d 2 x, x) = π 1 K k,d 2 x, x). [H k,d x, x) + H k 1,d x, x)] K k,d 2 x, x) c nk [1 ε)n],d 2 x, x) c n [1 ε)n] k=[1 λ)n] H k,d 2 x, x), where ε := 1 ρ)/d. Evdently, by nducton ths estmate yelds 2.20) for d odd. To establsh the result for d even, we only have to rove estmate 2.20) for d = 2. By the defnon of F d r, t), we have F 0 r, t) = e 1 r 1+r t2 = π 1/2 1 r 2 ) 1/2 F 1 r, t) ) 1/2 = π 1/2 1) r 2 h 2 kt)r k = π 1/2 n=0 k=0 where a 2 = 1) ) 1/2 and a2 1 = 0. Hence, usng 6.16) k a k h 2 x) r k, Consequently, k H k,2 x, x) H k 2,2 x, x) = π 1/2 a k h 2 t). H k,2 x, x)+h k 1,2 x, x) = π 1/2 A smle combnatoral formula shows that k a l = l=0 [k )/2] l=0 k l=0 l a l h 2 t) = k k h 2 t) a l. ) 1/2 1) l = Γ [ k 2 ]) l Γ 1 k 2 )Γ1 + [ 2 ]), whch s ostve for all 0 k. Furthermore, by Γk + a)/γk + 1) k a 1 t follows that k l=0 a l ck 1/2 for 0 αk for any α < 1. Therefore, αk H k,2 x, x) + H k 1,2 x, x) ck 1/2 h 2 t) and summng over k we get n k=[1 λ)n] H k,2 x, x) cn 1/2 K [ρn],1 t, t), l=0 whch establshes 2.20) for d = 2.

28 28 PENCHO PETRUSHEV AND YUAN XU 6.2. Proofs for Sectons 4-5. Proof of Lemma 4.7. Let 6.17) b x) := b η dx, R η )) σ, η X where dx, E) stands for the l dstance of x from E R d. Evdently, 6.18) b x) cb x) and b ξ1 Rξ x) cb x), x R d, ξ X. We wll show that 6.19) b x) cm s ω X b ω 1 Rω ) x), x R d. In vew of 6.18) ths mles 4.10)-4.11), and hence Lemma 4.7. By the constructon of the tles {R ξ } n 2.33)-2.34) t follows that there exsts a constant c > 0 deendng only on d such that Q := ξ X R ξ [ c 2, c 2 ] d. Fx x R d. To rove 6.19) we consder two cases for x. Case 1. x > 2c 2. Then dx, R η ) > x /2 for η X and hence b x) = b η dx, R η )) σ c 2 x ) σ b η η X η X 6.20) c4dλ 2 x ) σ b η s η X ) 1/s, where λ := 1 mn{1, 1/s} and for the last estmate we use Hölder s nequalty f s > 1 and the s-trangle nequalty f s < 1. Denote Q x := [ x, x ] d. Notce that Q Q x. From above we nfer b x) c4dλ x d/s 1 ) sdy ) 1/s 2 x ) σ b η 1 Rξ y) Q x Q x η X ) c2 2dλ σ) x d/s σ M s b η 1 Rξ )x) cm s b η 1 Rξ x) η X η X as clamed. Here we used the fact that σ d max{2, 1/s}. Case 2. x 2c 2. To make the argument more transarent we frst subdvde the tles {R η } η X nto boxes of almost equal sdes of length 2. By the constructon of the tles see 2.33)) there exsts a constant c > 0 such that the mnmum sde of each tle R η s c2. Now, evdently each tle R η can be subdvded nto a dsont unon of boxes R θ wth centers θ such that θ + [ c2 1, c2 1 ] R θ θ + [ c2, c2 ]. Denote by X the set of centers of all boxes obtaned by subdvdng the tles from X. Also, set b θ := b η f R θ R η. Evdently, 6.21) b x) := b η dx, R η )) σ η X θ X b θ dx, R θ )) σ

29 DECOMPOSITION OF SPACES INDUCED BY HERMITE EXPENSIONS 29 and 6.22) b η 1 Rη = b θ 1 Rθ. η X η X Denote Y 0 := {θ X : 2 θ x c}, Y m := {θ X : c2 m 1 2 θ x c2 m }, and Q m := {y R d : y x c2 m + 1)2 }, m 1. Clearly, #Y m c2 md, θ Ym R θ Q m, and X = m 0 Y m. Smlarly as n 6.20) b θ dx, R θ )) σ ) 1/s c2 mσ b θ c2 mσ 2 mdλ b θ s θ Y m θ Y m θ Y m c2 mσ dλ d/s) 1 ) sdy ) 1/s b θ 1 Rθ y) Q m Q m θ Y m ) c2 mσ d max{1,1/s}) M s b η 1 Rη x), η X where we used 6.22). Summng u over m 0, takng nto account that σ > d max{2, 1/s}), and also usng 6.21) we arrve at 6.19). Proof of Lemma 4.8. For ths roof we wll need an addtonal lemma. Lemma 6.4. Suose g V 4 and ξ X. Then for any k > 0 and L > 0 we have for x, x 2R ξ 6.23) gx ) gx ) c2 x x gη) ξ η ) k and 6.24) gx ) gx ) ĉ2 L x x η X η X gη) ξ η ) k, f ξ > 1+2δ)2 +1. Here c, ĉ deend on k, d, and δ, and ĉ deends on L as well; 2R ξ R d s the set obtaned by dlatng R ξ by a factor of 2 and wth the same center. Proof. Let Λ 4 be the kernel from 2.8) wth n = 4, where â s admssble of tye a) wth v := δ. Then Λ 4 g = g, snce g V 4, and Λ 4 x, ) V [1+δ)4 ]. Note that [1 + δ)4 ] + 4 2N 1. Therefore, we can use the cubature formula from Corollary 2.8 to obtan gx) = Λ 4 x, y)gy)dy = λ η Λ 4 x, η)gη), R d η X where the weghts λ ξ obey 2.35) see also 2.36)-2.37)). Hence, for x, x 2R ξ gx ) gx ) η X λ η Λ 4 x, η) Λ 4 x, η) gη) 6.25) c x x η X λ η su Λ 4 x, η) gη). x 2R ξ

30 30 PENCHO PETRUSHEV AND YUAN XU Note that 4[1 + δ)4 ] + k + 1) + 2) 1/2 1 + δ)2 +1 for suffcently large deendng on k and δ). Therefore, we have from 2.17)-2.18) 6.26) Λ 4 x, η) and for any L > 0 we need L k) 6.27) Λ 4 x, η) c2 d+1) x η ) k, x Rd, η X, c2 2L x η ) k, f x > 1+δ)2 +1 or η > 1+δ)2 +1. Suose ξ > 1 + 2δ)2 +1, then 2R ξ {x R d : x > 1 + δ)2 +1 } for suffcently large. Combnng 6.25) wth 6.27) and 2.37) we get 6.28) gx ) gx ) c2 d/3 x x su x 2R η X ξ 2 k+l) x η ) k gη), where we used that dam 2R ξ ) c2 /3. However, for any x 2R ξ we have ξ η ξ x + x η ) c2 /3 + x η ) c x η ). We use ths n 6.28) to obtan 6.24) for suffcently large. One roves 6.23) n a smlar fashon. In the case c estmates 6.23)-6.24) follow easly by 6.25). We are now reared to rove Lemma 4.8. Let g V 4. Pck l 1 suffcently large to be determned later on) and denote for ξ X 6.29) X +l ξ) := {η X +l : R η R ξ } and 6.30) d ξ := su{ gx ) gx ) : x, x R η for some η X +l ξ)}. We frst estmate d ξ, ξ X. Case A: ξ 1 + 3δ)2 +1. By 2.36) t follows that for suffcently large l deendng only on d and δ) η X+l ξ)r η 2R ξ. Hence, usng Lemma 6.4 see 6.23)) wth k σ, we get 6.31) d ξ c2 l η X gη) ξ η ) σ, for suffcently large deendng only on d and δ), where c > 0 s a constant ndeendent of l. Case B: ξ > 1+3δ)2 +1. By 2.36) x > 1+2δ)2 +1 for x η X+l ξ)r η f s suffcently large. We aly estmate 6.24) of Lemma 6.4 wth k σ and L = 1 to obtan 6.32) d ξ c2 gη) ξ η ) σ. η X To estmate M ξ, ξ X, we consder two cases for ξ. Case 1: ξ 1 + 4δ)2 +1. By 2.36), we have for suffcently large : 6.33) R ξ ξ + [ 2, 2 ] d, and R η η + [ 2 l, 2 l ] d, η X +l ξ).