Characterizations of function spaces on the sphere using frames

In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as $L^p$, $H^p$ and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case $\mathbb{R}^d$. We also study a related nonlinear $m$-term approximation problem on $\sph$. In particular, we prove both a Jackson--type inequality and a Bernstein--type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions'', {\it Amer. J. Math.} {\bf 114} (1992), no. 4, 737--785).


Introduction and summary of main results
1.1. Notations and basic facts. We start with some necessary notations. Given an integer d ≥ 3, we denote by S d−1 the unit sphere of the d-dimensional Euclidean space R d and dσ(x) the usual Lebesgue measure on S d−1 normalized by 2 FENG DAI parameters indicated as subscripts, and the notation A ≍ B means that there are two inessential positive constants C 1 , C 2 such that C 1 A ≤ B ≤ C 2 A.
For each nonnegative integer k, let H k denote the space of spherical harmonics of degree k on S d−1 and Y k the orthogonal projection of L 2 (S d−1 ) onto H k . As is well known, for f ∈ L 2 (S d−1 ), where for x = (x 1 , x 2 , . . . , x d ), y = (y 1 , y 2 , . . . , y d ) ∈ R d , x · y = x 1 y 1 + x 2 y 2 + . . . x d y d , and P d−2 2 k (t) denotes the usual ultraspherical polynomial of order d−2 2 normalized by P . ( For precise definition of ultraspherical polynomials, we refer to [Sz,p. 81].) Evidently, the formula (1.1) allows us to extend the definition of Y k (f ) to contain all distributions f ∈ S ′ (S d−1 ). For an integer n ≥ 0 we denote by Π n the space of all spherical polynomials of degree at most n on S d−1 (i.e., polynomials in d-variables of total degree at most n restricted to S d−1 ). It is well known that Π n = n k=0 H k and dim Π n = n k=0 dim H k ≍ n d−1 . For more information on spherical harmonics, we refer to [SW, Ch IV].
1.2. Construction of polynomial frames on S d−1 . Various nonstationary wavelets or frames have been constructed on the sphere by many authors ( see [ADS, AV, DDSW, FGS, G, MNPM, NW]). Although each of these wavelets or frames has its own advantages, to the best of our knowledge none of them has been shown useful in the characterizations of classic function spaces on the sphere. In this subsection, we will construct a polynomial frame on S d−1 , for which many function spaces on the sphere S d−1 , such as L p , H p and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions. Our construction is motivated by the approach taken in the pioneer work [MNPM], where the authors used positive cubature formulae to introduce a class of polynomial frames suitable for analyzing data on the sphere. It is based on the following theorem: Theorem A. There exists a constant γ > 0 depending only on d such that for any integer N > 0 and any finite set {ξ k } k∈Ω of distinct points ξ k ∈ S d−1 satisfying there exists a set of numbers 0 ≤ a N,k ≤ C d N −(d−1) , k ∈ Ω such that for any f ∈ Π N , 0 < p ≤ ∞ and t ≥ 0, where the constants of equivalence depend only on d and p when p is small, and in sequel, we employ the slight abuse of notation that 0 0 = 1. Theorem A under the restriction t ≤ min{p, 1} was obtained in [BD,Theorem 3.1]. It will be shown in Section 5 that the restriction t ≤ min{p, 1} is in fact not necessary.
An equality like (1.3) with nonnegative coefficients is called a positive cubature formula while an equivalence like (1.4) is called an Marcinkiewicz-Zygmund (MZ) type inequality. It should be pointed out that positive cubature formulae and MZ inequalities for 1 ≤ p ≤ ∞ based on function values at scattered sites on S d−1 were first established in the fundamental paper [MNW]. Now to each integer j > 0 we assign a finite set {x j,k : with γ as in Theorem A. Evidently, #Λ d j ≍ 2 j(d−1) and by Theorem A, there exists a set of numbers 0 ≤ λ j,k ≤ C d 2 −j(d−1) , k ∈ Λ d j such that for any 0 < p ≤ ∞, 0 ≤ t ≤ min{p, 1} and any f ∈ Π 2 j+4 , (1.5) where the constants of equivalence depend only on d and p when p is small. For convenience, we also set Λ d 0 = {0}, λ 0,0 = 1 and take x 0,0 to be any fixed point on S d−1 .
Let φ be a nonnegative C ∞ -function on R supported in {x ∈ R : 1 2 ≤ |x| ≤ 2} and satisfying Together with φ we define a sequence of functions and P k (t) is defined by (1.2). We index these functions by the spherical caps We also use the notation B j to denote the set of spherical caps B(x j,k , 2 −j π), ( k ∈ Λ d j ) and B to denote the union of the B j , j ≥ 0. Thus, associated with each distribution f on S d−1 , there is a series B∈B f, ψ B ψ B , and moreover, for each spherical polynomial f , we have with only a finite number of nonzero coefficients f, ψ B , as can be easily verified. We will keep the above notations for the rest of the paper.
We have three purposes in this paper. First, we want to study the unconditional convergence of the series B∈B f, ψ B ψ B (x). Second, we want to characterize classic function spaces on S d−1 , such as L p , H p and Besov spaces, using the coefficients f, ψ B , (B ∈ B). Third, we wish to investigate a related nonlinear m-term approximation problem. Our main results will be summarized in the next subsection.
1.3. Summary of main results. To state our results in a unified manner, we identify H ∞ (S d−1 ) with C(S d−1 ), and note that (see [C]) for 1 < p < ∞, the Hardy space H p (S d−1 ) coincides with the Lebesgue space L p (S d−1 ) and f p ≍ f H p with the constants of equivalence depending only on d and p.
Our first result gives a characterization of the spaces H p , 0 < p < ∞.
Theorem 1.1. If 0 < p < ∞ and f ∈ H p (S d−1 ), then B∈B f, ψ B ψ B converges unconditionally to f in the H p -metric, and moreover with the constants of equivalence depending only on d, p and φ. In addition, if 0 < p < ∞ and {a B } B∈B is a sequence of complex numbers such that ei- with the constants C 1 and C 2 depending only on d, p and φ.

CHARACTERIZATIONS OF FUNCTION SPACES ON THE SPHERE USING FRAMES 5
Our next result concerns a characterization of the Besov spaces. For α > 0 and 0 < p, τ ≤ ∞, we define and define the Besov space B α τ (H p ) to be a linear space of distributions on S d−1 endowed with the quasi-norm | · | B α τ (H p ) . We point out that Besov spaces on the sphere were introduced and investigated by Nikol'skii, Lizorkin and Rustamov in a series of papers ( see [R] and the references there). These spaces can be equivalently characterized using the K-functionals or moduli of smoothness on the sphere ( see [R]).
with the usual change when p = ∞ or τ = ∞, where the constants of equivalence depend only on α, τ, p and φ. In addition, if {a j,k : j = 1, 2, . . . , k ∈ Λ d j } is a sequence of complex numbers such that then the series ∞ j=0 k∈Λ d j a j,k ψ j,k converges unconditionally to some f ∈ B α τ (H p ) in the H p -metric and moreover, with the usual change when p = ∞ or τ = ∞.
Of particular interest are the spaces B α τ (H τ ), for which we have with the usual change when τ = ∞, on account of Theorem 1.2. We point out that for the usual Euclidean space R d , results similar to Theorems 1.1 and 1.2 can be found in [FJ], [HW,Chapter 7] and [DJP].
Finally, we state our results on nonlinear approximation. For 0 < p ≤ ∞, f ∈ H p and an integer n > 0, we denote by Γ n ≡ Γ n,f,p a set of n spherical caps B ∈ B such that min and define the greedy type algorithm G p n (f ) by Such an algorithm is well defined, as was shown in [T1, remark 1.1]. We refer to the impressive survey paper [T2] for the background information of greedy algorithm.
The inequality (1.13) is a direct theorem of approximation (the Jackson inequality) while the inequality (1.14) is an inverse theorem ( the Bernstein inequality). Once (1.13) and (1.14) are established, then by the standard method (see [DP]), the following characterization result holds for 0 < β < α: where f ∈ H p , 0 < p ≤ ∞ and τ = (β/(d−1)+1/p) −1 . For results on the nonlinear approximation associated with wavelet decomposition in L p (R d ) ( 0 < p ≤ ∞), we refer to [DJP], [DPY] and [Jia]. We organize this paper as follows. Section 2 contains three lemmas which will be used frequently in the proofs of our main results. We prove Theorem 1.1 in Section 3 for the case 1 < p < ∞, and in Section 4 for the case 0 < p ≤ 1. The proof of Theorem 1.2 and those of Theorems 1.3 and 1.4 are given in Sections 5 and 6 respectively.
Finally, we point out that our paper does not give any effective algorithms for spherical wavelets. For results in this direction, we refer the reader to the papers [SS1], [SS2] and [KL]. We also note that characterizations of function spaces using wavelets on stratified Lie groups had been considered in [L]. (The author would like to thank an anonymous referee very much for kindly pointing out these references ( [SS1], [SS2], [KL], [L]) to him.) For more recent work on spherical frames, we refer to the impressive paper [NPW] by F. J. Narcowich, P. Petrushev and J. D. Ward, and also the nice survey paper [MP] by H. N. Mhaskar and J. Prestin.

Three useful lemmas
This section contains three lemmas that will be useful in the proofs of the main results in this paper. The first two lemmas ( Lemmas 2.1 and 2.2) are in essence known, while the last one (Lemma 2.3) is new and will be of fundamental importance.
For the statement of Lemma 2.1, we define, for f ∈ S ′ (S d−1 ), where G j is defined by (1.8).
In addition, if f is a spherical polynomial, then with only a finite number of nonzero terms.
Proof. The inequality (2.2) is a simple consequence of the well-known Hörmandertype multiplier theorem for spherical harmonics (see [S,Theorem 2]), while the identity (2.3) follows directly from (1.7) and the definition.
To state our next lemma, we suppose ϕ is a C ∞ -function on [0, ∞) supported in [0, 2] and equal to a constant on [0, 1 2 ], and define with P k (t) as defined in (1.2). Then, with these notations, we have Lemma 2.2. For θ ∈ [0, π] and any positive integer ℓ, We note that for the usual Cesàro kernel and the order (N θ) −d on the right-hand side of this last inequality cannot be further improved. The significance point of Lemma 2.2 is that the positive integer ℓ can be chosen as big as we like, which will play a very important role when we deal with the case 0 < p < 1 in the later sections. The proof of Lemma 2.2 is contained in [BD,Lemma 3.3].
The following lemma will be of fundamental importance in the proofs of our main results.
3. Proof of Theorem 1.1 for 1 < p < ∞ We need the following lemma, which means that for each j ∈ Z + and k ∈ Λ d j the function ψ j,k is highly localized in the spherical cap B(x j,k , 2 −j−1 ).
Bernstein's inequality for trigonometric polynomials, we obtain that for t ∈ [0, 1 2 j+1 ], and the inequality g j,k (x) ≤ C d,φ |ψ j,k (x)| then follows. For the proof of the inequality we use Lemma 2.2 to obtain that In either case, we have the desired estimate (3.2), and the proof is therefore complete.
Proof of Theorem 1.1 for 1 < p < ∞. First, we show that for any sequence {a B } B∈B of complex numbers, Indeed, since 0 ≤ λ j,k ≤ C d 2 −j(d−1) , by Lemma 3.1 applied to r = 1 it follows that for any B ∈ B, |ψ B | ≤ C d,φ |B| − 1 2 M (χ B ), which together with the well-known Fefferman-Stein inequality implies the inequality (3.3).
Second, we show that for any sequence {a B } B∈B of complex numbers and any finite subset F ⊂ B, Once (3.4) is proved, then by a standard argument, we deduce that the series B∈B a B ψ B converges unconditionally in the space L p provided that B∈B |a B | 2 |ψ B | 2 1 2 ∈ L p . This together with (3.3) will imply the second assertion of Theorem 1.1 in the case 1 < p < ∞.
For the proof of (3.4), we define where σ j (f ) is defined by (2.1). For j ∈ Z + and k ∈ Λ d j , we write B j,k = B(x j,k , 2 −j π), B ′ j,k = B(x j,k , 2 −j−1 ), and we define a ′ j,k = a B j,k if B j,k ∈ F , and = 0 otherwise. Also, we set h = B∈F a B ψ B . Let g ∈ L p ′ be such that g p ′ = 1 and h p = Invoking Lemma 2.1, Lemma 2.3 with β = 1 and N = 2 j , and the Fefferman-Stein inequality, we deduce ∞ j=0 |σ * * j (g)| 2

CHARACTERIZATIONS OF FUNCTION SPACES ON THE SPHERE USING FRAMES 11
while using the first inequality in Lemma 3.1, we have The desired inequality (3.4) then follows. Finally, we show the first assertion of Theorem 1.1. We claim that it will suffice to prove that for f ∈ L p , In fact, once (3.6) is proved, then by the second assertion of Theorem 1.1 we just proved, it follows that for f ∈ L p , the series B∈B f, ψ B ψ B is convergent unconditionally in L p , and by the usual density argument we must have This together with (3.3), (3.4) and (3.6) will imply the desired equivalences and hence the first assertion of Theorem 1.1.
For the proof of (3.6), we recall that f, ψ j,k = λ j,k σ j (f )(x j,k ) and 0 ≤ λ j,k ≤ C d 2 −j(d−1) . Thus, by the definition, it's easy to verify that where σ * * j is defined by (3.5). The desired inequality (3.6) then follows by Lemma 2.1, Lemma 2.3 and the well-known Fefferman-Stein inequality. This completes the proof.
4. Proof of Theorem 1.1 for 0 < p ≤ 1 We start with some basic definitions and facts related to the Hardy spaces for all z ∈ B d . p z belongs to S(S d−1 ) and is called the Poisson Kernel. Given a distribution f ∈ S ′ (S d−1 ), we define its radial maximal function by and its H p -quasi-norm ( for a given 0 < p < ∞) by f H p = P + f p . For 0 < p < ∞, the Hardy space H p (S d−1 ) is defined to be the space of all distributions f ∈ S ′ (S d−1 ) with f H p < ∞. It is well known (see [C]) that if 1 < p < ∞ then the Hardy space H p coincides with the Lebesgue space L p and f p ≍ f H p with the constants of equivalence depending only p and d. We will restrict ourselves to the case 0 < p ≤ 1 for the rest of this section.

FENG DAI
For 0 < p ≤ 1, 1 ≤ q ≤ ∞ and a nonnegative integer s, a regular (p, q, s) -atom centered at a point y ∈ S d−1 is a function a in L q (S d−1 ) satisfying the following three conditions: (i) supp a ⊂ B(y, r) for some r > 0; An exceptional atom is a function a in L ∞ (S d−1 ) with a ∞ ≤ 1. Then the well-known atomic decomposition theorem (see [C,Proposition 3.1 Given 0 < p ≤ 1, a p-molecule centered at a point y ∈ S d−1 is a function m ∈ L 2 (S d−1 ) satisfying the following two conditions: (i ′ ) For some r > 0 and s > (d − 1)( 2 p − 1), According to [C, p. 234], for 0 < p ≤ 1, any p-molecule m must satisfy m H p ≤ C p,d .
For the proof of Theorem 1.1, we need the following where σ j is defined by (2.1).
|σ j (f )| 2 1 2 is bounded on L 2 , by the atomic decomposition theorem it will suffice to prove that for any regular (p, ∞, s)-atom a, For the proof of (4.1), we suppose a is a regular (p, ∞, s)-atom supported in B(x 0 , r) for some x 0 ∈ S d−1 and r ∈ (0, 1 4 ). Then using Hölder's inequality, we

CHARACTERIZATIONS OF FUNCTION SPACES ON THE SPHERE USING FRAMES
To prove this last inequality, we claim that for x ∈ S d−1 \B(x 0 , 4r) and ℓ = d+2s+2, . Once the claim (4.3) is proved, then by straightforward calculation, we deduce that for , from which the desired inequality (4.2) will follow. Now the proof of Lemma 4.1 is reduced to the proof of the claim (4.3). We recall that Hence, by the definition of regular (p, ∞, s)-atom it follows that For x ∈ S d−1 \ B(x 0 , 4r) and y ∈ B(x 0 , r), we write θ = d(x, x 0 ) and t = d(x, y). Then, evidently, 3θ 4 ≤ t ≤ 5θ 4 and |x · (y − x 0 )| = 2 sin θ−t 2 sin θ+t 2 ≤ 9 8 θr. Hence, by Lemma 2.2 it follows that for any ℓ > 0, x ∈ S d−1 \ B(x 0 , 4r) and y ∈ B(x 0 , r) where θ = d(x, x 0 ). Substituting this last estimate into (4.4), we obtain that for proving the claim (4.3). This completes the proof.
Proof of Theorem 1.1 for 0 < p ≤ 1. Following the proof in the last section, we need only verify the following three assertions in the case 0 < p ≤ 1: (a) For any sequence {a B } B∈B of complex numbers, (c) for any finite subset F ⊂ B and any sequence {a B } B∈F of complex numbers, Assertion (a) follows directly from the Fefferman-Stein inequality and the second inequality in Lemma 3.1 with 0 < r < p.
For the proof of assertion (b), we define for β > 0 and j ∈ Z + , Then by Lemma 2.3 applied to N = 2 j , we have and hence, by Lemma 4.1 and the Fefferman-Stein inequality, we obtain that for Now assertion (b) follows from (4.5) and the following inequality, which can be easily verified: It remains to prove assertion (c). For simplicity, we define a B = 0 for B ∈ B \ F , we denote by D the set of all spherical caps B(x j,k , 2 −j−1 ), (j ∈ Z + , k ∈ Λ d j ), for each j ∈ Z + , k ∈ Λ d j , we write a j,k = a B(x j,k ,2 −j−1 ) := a B(x j,k ,2 −j π) , λ B(x j,k ,2 −j−1 ) := λ j,k , ψ B(x j,k ,2 −j−1 ) := ψ j,k , and for c > 0, B = B(x, r), we write cB := B(x, cr).
First, we observe that for 0 < p ≤ 1 p since any two quasi-norms on a finite-dimensional linear space are equivalent. Thus, without loss of generality, we may assume (4.6) a j,k = 0, for all 2 j ≤ 8(d − 1)( 1 p − 1) and k ∈ Λ d j .
Then by the first inequality in Lemma For the proof of this last inequality, for an integer k ∈ Z we let Then summation by parts yields Then it's easily seen that B∈D a B ψ B = k∈Z i∈Γ k b(k, i)m k,i . Therefore, for the proof of (4.7), it is sufficient to show that each m k,i is a p-molecule up to an absolute constant and (4.9) Since the spaces {H k } ∞ k=0 (of spherical harmonics) are mutually orthogonal, it follows by the assumption (4.6) that Thus, to show that m k,i is a p-molecule up to an absolute constant, it suffices to prove that for any s > 0, (4.10) where z ≡ z k,i denotes the center of B i k , r ≡ r k,i denotes the radius of B i k . In fact, since However, on the other hand, for each B ∈ D k with B ⊂ 2B i k , and each x ∈ S d−1 \ 4B i k , applying Lemma 2.2, we obtain Hence, We then deduce by a straightforward calculation that which together with (4.11) implies (4.10). It remains to prove (4.9). We observe that Thus, by the definition, since {B i k } i∈Γ k is a sequence of mutually disjoint subsets of Ω k . This combined with (4.8) gives (4.9) and therefore completes the proof.

Proof of Theorem 1.2
Recall that σ j is defined by (2.1). We need the following Lemma 5.1. For α > 0 and 0 < p, τ ≤ ∞, with the usual change when τ = ∞, where the constants of equivalence depend only on p, d, α, τ and φ.
Proof. By the definition, it's easily seen that the series ∞ j=0 σ j • σ j (f ) converges to f in the space H p and for each k ∈ Z + , k j=0 σ j • σ j (f ) ∈ Π 2 k −1 . Thus, for each k ∈ Z + and f ∈ H p , where q = min{p, 1}. Since the operators σ j , j ∈ Z + are uniformly bounded on H p , by the definition and Hardy-type inequality it follows that with the usual change when τ = ∞.
On the other hand, noting that σ j (g) = 0 for any g ∈ Π [2 j−2 ] and j ≥ 1, we obtain that for j ≥ 1 This together with the uniform boundedness of the operators σ j on H p implies the desired inverse inequalities with the usual change when τ = ∞. This completes the proof.
As indicated in Section 1, the MZ-type inequality (1.4) in Theorem A under the restriction 0 ≤ t ≤ min{p, 1} was proved in [BD,Theorem 3.1]. Our Lemma 5.2 below asserts that this same inequality, in fact, holds for t > min{p, 1} as well.
Lemma 5.2. Suppose that {ξ k } k∈Ω is a finite subset of S d−1 and {a N,k } k∈Ω is a sequence of nonnegative numbers smaller than C d N −(d−1) . Suppose further that the MZ inequality (1.4) holds for all f ∈ Π N and 0 ≤ t ≤ min{p, 1}. Then we have, for any 0 < p ≤ ∞, any f ∈ Π N and all α ≥ 0, Then by Theorem 1.2, we have Ei |g(x)| τ dσ(x) ≤ C p,α n 1− τ p g τ p = C p,α n