Applied and Computational Harmonic Analysis

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1 Appl Comput Harmon Anal 32 (2012) Contents lists available at ScienceDirect Applied and Computational Harmonic Analysis wwwelseviercom/locate/acha Multiscale approximation for functions in arbitrary Sobolev spaces by scaled radial basis functions on the unit sphere QT Le Gia a,, IH Sloan a,hwendland b a School of Mathematics and Statistics, University of New South Wales, Sydney, Australia b Mathematical Institute, Oxford University, UK article info abstract Article history: Received 18 April 2011 Revised 25 July 2011 Accepted 30 July 2011 Available online 5 August 2011 Communicated by Charles K Chui In this paper, we prove convergence results for multiscale approximation using compactly supported radial basis functions restricted to the unit sphere, for target functions outside the reproducing kernel Hilbert space of the employed kernel 2011 Elsevier Inc All rights reserved Keywords: Multiscale approximation Radial basis function Unit sphere 1 Introduction In the geosciences data are often collected at scattered sites on the unit sphere Moreover, the geophysical data typically occur at many length scales: for example, the topography of central Australia varies slowly, while that of the Himalayan mountains varies rapidly To handle multiscale data at scattered locations, we consider an approximation scheme based on radial basis functions of different scales, which are generated from a single underlying radial basis function (RBF) Φ using a sequence of scales 1, 2, with limit zero, with the scale becoming smaller as the point set becomes denser The scaled RBF is defined by Φ = c Φ( ), which is then restricted to the unit sphere There is significant difficulty in dealing with more than one scale at the same time, namely that the associated reproducing kernel Hilbert space (RKHS) or native space has a different inner product for each scale For this reason a multiresolution analysis within a single Hilbert space, of the kind familiar from wavelet analysis, does not seem possible for scaled RBFs on either a sphere or a Euclidean region In a recent paper [1], we resolved that issue, and constructed a multiresolution analysis for sufficiently smooth functions on the unit sphere S n based on RBFs of different scales We have proved a convergence result for a multiscale approximation algorithm for functions from the RKHS of the given compactly supported kernel, if the RKHS is a sufficiently smooth Sobolev space In this paper, we shall extend our convergence results to rougher target functions Specifically, if the employed kernel is the reproducing kernel of, say, H σ (S n ) then the new results will hold for all target functions from H β (S n ) with σ >β>n/2 The paper is organized as follows In Section 2, after reviewing the necessary background on spherical harmonics and spherical kernels defined from radial basis functions, we review Sobolev splines and prove a new result on the asymptotic behavior of the Fourier Legendre coefficients of spherical kernels defined by restricting the Sobolev splines to the unit * Corresponding author addresses: qlegia@unsweduau (QT Le Gia), isloan@unsweduau (IH Sloan), HolgerWendland@mathsoxacuk (H Wendland) /$ see front matter 2011 Elsevier Inc All rights reserved doi:101016/acha

2 402 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) sphere In Section 3 we review the method of multiscale approximation using scaled compactly supported radial basis functions which was analyzed in [1] for functions lying in the reproducing kernel Hilbert space defined by the spherical kernel In Section 4 we give a convergence analysis for rougher target functions The main result of the paper is stated in Theorem 45 2 Preliminaries 21 Positive definite bizonal kernels on the unit sphere The unit sphere S n is defined by S n ={x R n+1 : x =1}, where x denotes the Euclidean norm of x Bizonal kernels defined on S n S n are kernels that can be represented as g(x, y) = g(x y) for all x, y S n, where g(t) is a continuous function on [ 1, 1] We shall be concerned exclusively with bizonal kernels of the type g(x, y) = a l P l (n + 1; x y), a l > 0, a l <, (1) where {P l (n + 1; t)} isthesequenceof(n + 1)-dimensional Legendre polynomials normalized to P l(n + 1; 1) = 1 Thanks to the seminal work of Schoenberg [2] and the later work of [3], we know that such a g is (strictly) positive definite on S n, that is, the matrix A := [g(x i, x )] M i, =1 is positive definite for every set of distinct points {x 1,,x M } on S n and every positive integer M For mathematical analysis, sometimes it is convenient to expand the kernel g into a series of spherical harmonics A detailed discussion on spherical harmonics can be found in [4] In brief, spherical harmonics are the restriction to S n of homogeneous polynomials Y in R n+1 which satisfy Y = 0, where is the Laplacian operator in R n+1 The space of all spherical harmonics of degree l on S n,denotedbyh l, has an orthonormal basis { Ylk : k = 1,, }, where N(n, 0) = 1 and = (2l + n 1)Γ (l + n 1) Γ(l+ 1)Γ (n) for l 1 The space of spherical harmonics of degree L will be denoted by P L := L H l ; it has dimension N(n + 1, L) Every function f L 2 (S n ) can be expanded in terms of spherical harmonics, f = ˆflk Y lk, ˆflk = S n f Y lk ds, where ds is the surface measure of the unit sphere Using the addition theorem for spherical harmonics (see, for example, [4, p 18]), we can write g(x, y) = ĝ(l)y lk (x)y lk (y), with ĝ(l) = ω n a l, (2) where ω n = S n ds is the surface area of S n We shall refer to ĝ(l) as the Fourier symbol of g 22 Kernels defined from radial basis functions Let Φ = ρ( ) : R n+1 R be a compactly supported radial basis function (RBF) with associated RKHS H σ +1/2 (R n+1 ) with σ > n/2 Examples of such RBFs are Wendland s functions (see [5]) By restricting the function Φ to the unit sphere S n R n+1, we have a positive definite, bizonal kernel on the unit sphere φ(x, y) = Φ(x y) = ρ ( x y ), x, y S n It is known (see [6]) that the Fourier symbol ˆφ(l) satisfies ˆφ(l) (1 + l) 2σ, ie there exist constants c 1, c 2 > 0such that c 1 (1 + l) 2σ ˆφ(l) c 2 (1 + l) 2σ, l 0 (3) In the remainder of the paper, we use c 1, c 2, to denote specific constants while c, C are generic constants that may take different values at each occurrence

3 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) The reproducing kernel Hilbert space (RKHS) of the kernel φ is given by { N φ := f D ( S n) : f 2 φ := ˆf } lk 2 φ(l) <, where D (S n ) is the set of all tempered distributions defined on S n Alternatively, N φ is the completion of span{φ(, x): x S n } with respect to the norm φ The norm obviously comes from the following inner product ( f, g) φ = ˆflk ĝ lk ˆφ(l), f, g N φ The kernel φ has the reproducing property with respect to this inner product, that is f (x) = ( f,φ(, x) ) φ, x Sn, f N φ The Sobolev space H σ (S n ) with real parameter σ is defined by H σ ( { S n) := f D ( S n) } : f 2 H σ (S n ) := (1 + l) 2σ ˆflk 2 < For σ = 0 the space is ust L 2 (S n ) Under the condition (3), the norms φ and H σ (S n ) are equivalent, since c 1/2 1 f φ f H σ (S n ) c 1/2 2 f φ For a given >0, we define the scaled version φ of the kernel φ by (4) φ (x, y) = n Φ ( (x y)/ ) = n ρ ( x y / ) (5) We can expand φ into a series of spherical harmonics φ (x, y) = φ (l)y lk (x)y lk (y), in which the Fourier coefficients satisfy the following condition (see [1, Theorem 62]) (6) c 1 (1 + l) 2σ φ (l) c 2 (1 + l) 2σ, l 0, (7) with the coefficients c 1 and c 2 from (3) possibly relaxed so that (7) holds for all 0 < 1 For a function f H σ (S n ), we define the norm corresponding to the scaled kernel φ by f φ = ( ˆf lk 2 φ (l) ) 1/2, and the corresponding inner product is ( f, g) φ = ˆflk ĝ lk ˆφ (l), f, g N φ Clearly the norms φ for different are all equivalent From (7) and (8) it follows (see [1, Lemma 31]) that c 1/2 1 g φ g H σ (S n ) c 1/2 2 n g φ (8) (9) (10) 23 Sobolev splines and associated spherical kernels Our target function f will be assumed to belong to the Sobolev space H β (S n ) for some β satisfying σ >β>n/2 We need an explicit formula for the reproducing kernel of H β (S n ) This will be defined via a Sobolev spline in H β+1/2 (R n+1 ) For ν > 0, the Sobolev spline (or Matérn function) is defined by S ν (x) = (2π) (n+1)/2 x ν K ν ( x ) 2 ν+(n 1)/2 Γ(ν + (n + 1)/2), x Rn+1,

4 404 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) where K ν is the K -Bessel function of order ν With the Fourier transform of a function f L 1 (R n+1 ) defined by ˆf (ω) = (2π) (n+1)/2 f (x)e ix ω dx, R n+1 the Fourier transform of S ν is given by [5, Theorem 613] Ŝ ν (ω) = (2π) (n+1)/2( 1 + ω 2) ν (n+1)/2 By setting ν = β n/2, the kernel Ψ(x y) := S ν (x y) is hence the reproducing kernel of the Sobolev space H β+1/2 (R n+1 ) with respect to the inner product ( f, g) H β+1/2 (R n+1 ) = ( ˆf ) (ω)ĝ(ω) 1 + ω 2 β+1/2 dω R n+1 For 0 < 1, we define the scaled version of Ψ by Ψ (x) = n Ψ(x/), x R n+1 (11) Similarly as before, we define the kernel ψ and its scaled version by restricting Ψ on the sphere, ψ(x, y) = Ψ(x y), ψ (x, y) = Ψ (x y), x, y S n (12) They can be expanded into a series of spherical harmonics as ψ (x, y) = ˆψ (l)y lk (x)y lk (y) We define the following norm ( f ψ := ˆf ) 1/2 lk 2 ˆψ (l) Then the RKHS of the kernel ψ is defined by (13) (14) N ψ := { f D ( S n) : f ψ < } The following two lemmas will give more information about this norm Lemma 21 Let Ψ be the reproducing kernel of H β+1/2 (R n+1 ),andletψ be defined by restricting Ψ to the unit sphere as in (12) Then the Fourier symbol ˆψ (l) satisfies c 3 (1 + l) 2β ˆψ (l) c 4 (1 + l) 2β, l 0, (15) where c 3, c 4 are two positive constants independent of and l Proof The proof essentially follows the proof of [1, Theorem 62] except for one point, namely the proof that ˆψ (l) c, for l 0, where c > 0 is independent of l and The corresponding proof in [1] was given under the assumption that Ψ is compactly supported with support in the unit ball, which is not the case here A proof for the present definition of Ψ is as follows We have ˆψ (l) = ω n Ψ ( 2 2t)P l (n + 1; t) ( 1 t 2) (n 2)/2 dt 1 ( ) ν ( ) 2 2t 2 2t = C n,ν n K ν P l (n + 1; t) ( 1 t 2) (n 2)/2 dt, 1 where C n,ν := ω n 1 2 ν (n 1)/2 (2π) (n+1)/2 Using the substitution 2 2t r =

5 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) and the bound 1 t 2 = (1 + t)(1 t) 2(1 t) = 2 r 2, together with P l (n + 1; t) 1 and K ν (r)>0forr > 0 and ν R, we have an upper bound 2/ ˆψ (l) C n,ν r ν K ν (r)r n 1 dr 0 C n,ν r ν K ν (r)r n 1 dr 0 = C n,ν 2 ν+n 2 Γ(n/2)Γ (ν + n/2), where in the last step we have used [7, Formula 11422, p 486] For the remainder of the proof we refer to [1, Theorem 62] 3 Multiscale approximation on S n Let X ={x 1,,x N } S n be a finite set of distinct points in S n Wedefinethemeshnormh X and the separation radius q X of this point set by h X = sup x S n min x X θ(x, x ), q X = 1 2 min θ(x i, x ), i where θ(x, y) = cos 1 (x y) is the geodesic distance on S n We define the interpolation operator I X, associated with the set X and the kernel φ by I X, f (x) = N b m φ (x, x m ), I X, f (x m ) = f (x m ) for all x m X (16) m=1 We have previously obtained in [1] the following result, which extends results obtained for = 1 in[8] to general (0, 1] Theorem 31 (See [1, Theorem 32]) Let h X be the mesh norm of the finite set X S n,andletg H σ (S n ) vanish on X Then there exists c 5 > 0 such that, for sufficiently small h X and all 0 < 1, g L2 (S n ) c 5 ( h X and, for f H σ (S n ), ) σ g φ, f I X, f L2 (S n ) c 5 ( h X ) σ f φ (17) (18) Suppose X 1, X 2, S n is a sequence of point sets with mesh norms h 1, h 2, respectively The mesh norms are assumed to satisfy h +1 μh for some fixed μ (0, 1) Let 1, 2, be a decreasing sequence of positive real numbers defined by = νh for some ν > 0 Taking the scale proportional to the mesh norm is desirable for both numerical stability and efficiency, since the sparsity of the interpolation matrix is maintained For every = 1, 2, we define the scaled SBF φ := φ We start with a widely spread set of points and use a basis function with scale 1 to recover the global behavior of the function f by computing f 1 = s 1 := I X1, 1 f The error, or residual, at the first step is e 1 = f f 1 Toreducetheerror, atthenextstepweuseafinersetofpointsx 2 and a finer scale 2, and compute a correction s 2 = I X2, 2 e 1 and a new approximation f 2 = f 1 + s 2, so that the new residual is f f 2 = e 1 I X2, 2 e 1 ; and so on Stated as an algorithm, we set f 0 = 0 and e 0 = f and compute for = 1, 2,, s = I X, e 1, f = f 1 + s, e = e 1 s Note that the algorithm makes f + e independent of, from which it follows that f + e = f 0 + e 0 = f, allowing us to identify e = f f as the error (or the residual) at stage, and f as the approximation to f at the stage Since f = s i = i=1 I Xi, i e i 1, i=1

6 406 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) we see that the approximation f is a linear combination of spherical basis functions at all scales up to level Wecanthink of s as adding additional detail to the approximation f 1 to produce f Also,sincee X = 0, it follows that f = f e interpolates f on X,or f X = f X The next theorem is the convergence result of [1] for functions from the associated reproducing kernel Hilbert space Theorem 32 (See [1, Theorem 41]) Let X 1, X 2, be a sequence of point sets on S n with mesh norms h 1, h 2, satisfying cμh h +1 μh for = 1, 2, with fixed μ, c (0, 1) and h 1 sufficiently small Let φ = φ be a kernel satisfying (7) with scale factor = νh and ν satisfying 1/h 1 ν β/μ with a fixed β>0 Assume that the target function f belongs to H σ (S n ) Then there is a constant C = C(β) independent of and f such that e φ +1 e 1 φ for = 1, 2,, with = Cμ σ, and hence there exists c > 0 such that (19) f f k L2 (S n ) c k f H σ (S n ) for k = 1, 2, Thus the multiscale approximation f k converges linearly to f in the L 2 norm if μ < C 1/σ 4 Approximation of rough target functions In this section, we analyze the multistage approximation method of the last section for target functions outside the RKHS N φ Hence, we prove convergence of the multiscale approximation scheme under the following conditions The interpolants are computed using the kernel φ, which generates H σ (S n ) with σ > n/2 The target function f, however, belongs only to H β (S n ) for some σ >β>n/2 41 A convergence theorem We need to collect a few auxiliary results before stating the convergence theorem for the multiscale approximation of rougher target functions Lemma 41 For 0 < 1 and all g H β (S n ) we have c 1/2 3 g ψ g H β (S n ) c 1/2 4 β g ψ Proof This follows from Lemma 21 The proof is essentially that of [1, Lemma 31] with σ replaced by β Lemma 42 (See [8, Theorem 33]) Let τ > n/2 and let X be a finite set in S n with sufficiently small mesh norm h X Then,thereisa constant C > 0, independent of X, such that for all f H τ (S n ) satisfying f X = 0, there holds f H γ (S n ) Ch τ γ X f H τ (S n ), 0 γ τ We also need a version of [9, Theorem 51], which asserts that for a sufficiently smooth function f there exists a spherical polynomial of degree at most L that interpolates f on a set X S n of distinct, scattered points and, simultaneously, is a good approximant to f We need a weak version of such a result for the Sobolev spaces generated by our scaled kernel ψ Lemma 43 Let f H β (S n ), β>n/2 and let X be a finite subset of S n with separation radius q X Let (0, 1] be given There exists a constant κ, which depends only on n and β, suchthatifl κ max{/q X, 1/}, then there is a spherical polynomial p P L such that p X = f X and f p ψ 5 f ψ We defer the proof of this lemma until the next subsection In the next lemma the notation is as in Theorem 32 with the addition that q = q X This lemma makes essential use of Lemma 43 in its proof Lemma 44 Let q h c q q and q Letψ := ψ,andlet f H β (S n ) with σ >β>n/2 Then there exists a constant C > 0, independent of X,suchthat e H β (S n ) C β e 1 ψ

7 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) Proof Choose L = κmax{ /q, 1/ } Then by Lemma 43, there is a spherical polynomial p P L that interpolates and approximates e 1,iep X = e 1 X and p e 1 ψ 5 e 1 ψ Wecanusep to split the error, e H β (S n ) = e 1 I X, e 1 H β (S n ) e 1 p H β (S n ) + p I X, e 1 H β (S n ) (20) The first term of (20) can be bounded using Lemmas 41 and 43, e 1 p H β (S n ) c 1/2 4 β e 1 p ψ 5c 1/2 4 β e 1 ψ Since p X = e 1 X the interpolant I X, e 1 is identical to I X, p Therefore the second term of (20) can be estimated via Lemma 42 and (10), p I X, e 1 H β (S n ) = p I X, p H β (S n ) Ch σ β p I X, p H σ (S n ) Ch σ β σ p I X, p φ Ch σ β σ p φ, where in the last step we used the fact that I X, p is the orthogonal proection with respect to the inner product (, ) φ onto span{φ (, x) : x X } Since p P L, 1 and β>σ, using the definition of the norm φ given by (8) and the condition (7) we have L p 2 φ 1 c 1 (1 + l) 2σ p l,k 2 1 L (1 + L) 2(σ β) c 1 C( /q ) 2(σ β) p 2 ψ, (1 + l) 2β p l,k 2 where in the last step we used the property that a 2a for a 1 Thus, combining these above estimates together with the fact that p ψ 6 e 1 ψ we obtain p I X, p H β (S n ) C β (h /q ) σ β e 1 ψ C β e 1 ψ In the following, we are going to prove that the multilevel algorithm also converges for target functions from H β (S n ) The proof generally follows the line of Theorem 41 in [1] with φ replaced by ψ but a new condition (see (ii) below) is needed, for now we need to use Lemma 44, which in turn rests on Lemma 43 Theorem 45 Let X 1, X 2,be a sequence of point sets in S n with mesh norms h 1, h 2,and separation radii q 1, q 2,satisfying (i) cμh h +1 μh for = 1, 2,with μ, c (0, 1), (ii) q h c q q for = 1, 2,with c q > 1 Let φ be a kernel generating H σ (S n ),iethefouriersymbolofφ satisfies (3), andletφ := φ be defined by (5) with scale factor = νh where 1/h 1 ν γ /μ 1 with a fixed γ > 0 Letψ be a kernel generating H β (S n ) with σ >β>n/2 and let ψ be the scaled version using the same scale factor Assume that the target function f belongs to H β (S n ) Then, there exists a constant C = C(γ )>0 such that, with = Cμ β, e ψ +1 e 1 ψ for = 1, 2, 3, and hence there exists a constant c > 0 so that f f k L2 (S n ) c k f H β (S n ) for k = 1, 2, 3, Proof We start with, using Lemma 21, e 2 ψ +1 1 c 3 ê,lk 2 ( l) 2β =: 1 c 3 (S 1 + S 2 ),

8 408 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) with S 1 := S 2 := l 1/ +1 l>1/ +1 ê,lk 2 ( l) 2β, ê,lk 2 ( l) 2β For the first sum S 1,since +1 l 1, S 1 l 1/ +1 ê,lk 2 2 2β 2 2β e 2 L 2 (S n ) Since e vanishes on X, by Lemma 42 we have e L2 (S n ) Ch β e H β (S n ) Combining this with Lemma 44, which gives e H β (S n ) C β e 1 ψ,wehave ( ) 2β h e 2 L 2 (S n ) C e 1 2 ψ Cμ 2β e 1 2 ψ For the sum S 2,since +1 l>1wehave, ( l) 2β <(2 +1 l) 2β (2 +1 ) 2β (1 + l) 2β So S 2 l>1/ +1 (2 +1 ) 2β ê,lk 2 (1 + l) 2β (2 +1 ) 2β e 2 H β (S n ) Using Lemma 44 again, which gives e H β (S n ) C β e 1 ψ,weobtain ( ) 2β +1 S 2 C e 1 2 ψ Cμ 2β e 1 2 ψ Putting all the estimates together, we obtain e ψ +1 Cμ β e 1 ψ e 1 ψ, which is the first statement of the theorem Finally, since e k = f f k vanishes on X k, we have by Lemma 42 and Lemma 41 that f f k L2 = e k L2 Ch β k e k H β (S n ) Ch β k β k+1 e k ψk+1 C e k ψk+1 C e k 1 ψk, since h k / k+1 = h k /(νh k+1 ) 1/(cγ ) Now we can apply the argument k times to obtain, with the aid of Lemma 41, f f k L2 C k e 0 ψk+1 = C k f ψk+1 C k f H β (S n ) 42 Existence of a polynomial interpolant We are now going to prove Lemma 43 To this end, we have to collect a number of auxiliary results, which are mainly adaptation of results from [9] The idea there is to lift the norms involving spherical basis functions (SBFs) on S n to those for RBFs on R n+1 We will once again work with the bizonal kernel ψ(x, y) = Ψ(x y) for x, y S n where Ψ is the reproducing kernel of the Sobolev space H β+1/2 (R n+1 ) For a given τ > 0, we define the function Ψ(τ ; ) via its Fourier transform, { Ψ(ω), ω τ, Ψ(τ ; ω) := (21) 0, ω > τ

9 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) Hence, Ψ(τ ; ) is given by Ψ(τ ; x) := (2π) (n+1)/2 Ψ(τ ; ω)e ix ω dω R n+1 = (2π) (n+1)/2 ω τ Ψ(ω)e ix ω dω A scaled version of Ψ(τ ; x) is defined by Ψ (τ ; x) := n Ψ(τ ; x/) (22) Consequently, the Fourier transform of Ψ (τ ; ) scales as { Ψ Ψ (τ ; ω) = Ψ(τ (ω) = Ψ(ω), ω τ, ; ω) = 0, ω > τ The bizonal kernel ψ ;τ (x, y) := Ψ (τ ; x y) x,y S n is an SBF with Fourier-Legendre coefficients given by (see [6, Theorem 41 and Corollary 43]) τ ˆψ ;τ (l) = (2π) (n+1)/2 0 t Ψ (t) J 2 l+(n 1)/2 (t) dt Besides this, we will also need the truncation of ψ to P L given by ψ L (x, y) := L ˆψ (l)y l,k (x)y l,k (y) The following result is [9, Lemma 42], with Ψ and ψ being replaced by Ψ and ψ, respectively (23) Proposition 46 Let X ={x 1,,x N } be a set of scattered points on S n and R N LetΨ be the reproducing kernel for H β+1/2 (R n+1 ) and ψ be its associated spherical kernel Let the scale parameter (0, 1] be given Then N N ψ (, x ) = Ψ ( x ) (24) =1 ψ =1 Ψ If there is a constant c > 0 such that when τ cl we have ˆψ ;τ (l) 1 ψ 2 (l) for all l>l, then we have 2 N ( ) ψ ψ L N ( (, x ) 2 Ψ (x x ) Ψ (τ ; x ) ) 2 (25) =1 ψ =1 Ψ We are now going to show that there is indeed such a constant c > 0 such that when τ cl we have ˆψ ;τ (l) 1 2 ˆψ (l) for all l>l For this, we need the following lemma Lemma 47 Let Ψ L 1 (R n+1 ) be radial and nonnegative, and let l 0If0 < 1,τ > 0 and ν = l + (n 1)/2, then τ ˆψ ;τ (l) = (2π) (n+1)/2 0 t Ψ (t) Jν 2 (t) dt 2(2π)(n+1)/2 e n Ψ(0)(eπ) n ( ω n (ν + 1) n eτ 2(ν + 1) ) 2ν n+1 Proof This follows from [9, Lemma 43] by replacing Ψ with Ψ and using Ψ (0) = n Ψ(0) Since ψ is a reproducing kernel for H β (S n ) with β>n/2, condition (15) and Lemma 47 yield ( ) ˆψ ;τ (l) eτ 2l ˆψ (l) C β,n(1 + l) 2β (l) n 2l

10 410 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) If we now set l 0 = 1/ then we have for all l l 0 that ( ) ˆψ ;τ (l) eτ 2l ˆψ (l) C β,n2 2β (l) 2β n 2l From this we can conclude that there is an L 0 l 0 such that for all l L L 0 and τ e 1 L we have ˆψ ;τ (l) ˆψ (l) C β,n2 2β (l) 2β n 2 2l 1 2 (26) Thus the conclusions of Proposition 46 hold We now state a proposition, whose proof can be found in [10, Theorem 21] or [11, Prop 31] It provides the theoretical framework for proving the existence of an interpolant that is also a good approximant Proposition 48 Let Y be a (possibly complex) Banach space, V a subspace of Y, andz a finite-dimensional subspace of Y,the dual of Y Ifthereisaγ > 1 such that for every z Z, z Y γ z V V, (27) then for y Y there exists v V such that v interpolates y on Z ; that is, z (y) = z (v) for all z Z In addition, v approximates y in the sense that y v Y (1 + 2γ )dist(y, V) We will apply this result to the case in which the underlying space is the RKHS of the scaled kernel ψ, that is Y = N ψ We will take Z = span{ x } N =1, where the points are distinct and come from a finite set X Sn Finally, we take V = P L, the span of the spherical harmonics of degree l L, withl at the moment fixed The quantities in Proposition 48 can be put in terms of the reproducing kernel ψ First, we observe that N N x ψ (, x ) =1 N =1 ψ Second, we have ψ = N N x = ψ =1 PL P L (, x ), =1 L ψ see formula (53) in [9] Moreover, since ψ L(, x ) and ψ (, x k ) ψ L(, x k) are orthogonal in N ψ Pythagorean theorem to obtain N ψ L (, x ) =1 2 ψ = N ψ (, x ) =1 2 ψ N [ ψ (, x ) ψ L (, x ) ] 2 =1 ψ for all, k, wecanusethe From this and the quantities above, it follows that Proposition 48 applied to our situation yields the following: if we can find a γ > 1 such that for every linear functional z = x we have N =1 [ψ (, x ) ψ L(, x )] 2 ψ N 1 1 =1 ψ (, x ) 2 γ, 2 (28) ψ then for f N ψ there exists a polynomial p so that f (x ) = p(x ) for 1 N In addition, p approximates f in the sense that f p ψ (1 + 2γ ) f ψ An explicit value of γ will be given in the proof of Lemma 43, as follows Proof of Lemma 43 We have, by using Proposition 46 and (26) that N =1 [ψ (, x ) ψ L(, x )] 2 ψ 2 N N =1 [Ψ ( x ) Ψ (τ ; x )] 2 Ψ =1 ψ (, x ) 2 ψ N, =1 Ψ ( x ) 2 Ψ

11 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) where we take τ = e 1 L and L L 0 as required for (26) Using the reproducing property of Ψ,wehave N [ Ψ ( x ) Ψ (τ ; x ) ] 2 N [ = k Ψ (x x k ) Ψ (τ ; x x k ) ] =1 Ψ, N ( ) ( x = n x k k [Ψ Ψ τ ; x )] x k, Similarly, using the reproducing property of Ψ again, we have 2 N N N ( ) x Ψ ( x ) = k Ψ (x x k ) = n x k k Ψ =1 Ψ,, By using an estimate from the proof of [12, Lemma 33] on the scaled data set Y = X/, we obtain the following estimate N [ ( x, k Ψ x k ) ( x Ψ τ ; x k )] N, k Ψ ( x x k ) C(τ Q X /) n+1 2(β+1/2), where C = C(β, n + 1) and Q X is the Euclidean separation radius for X as a subset of R n+1 For any discrete set X, we know that Q X is comparable to q X,indeedq X Q X (2/π)q X Combining all of the results and using τ = e 1 L yields N =1 [ψ (, x ) ψ L (, x )] 2 ψ N =1 ψ (, x ) 2 ψ C (Lq X /) n 2β, where C = 2 n+1 2β C/(eπ) n 2β We can choose κ > 0 such that C κ n 2β 3 Sinceβ>n/2 wehaven 2β <0, thus, if 4 Lq X / κ, thenc (Lq X /) n 2β 3 4 Moreover, by enlarging κ if necessary, we can also ensure that the condition L L 0 required for (26) is satisfied This means that (28) holds with γ = 2 provided that L κ max{q X /, 1/} Hence, we have shown that we can apply Proposition 48 in this situation, which finishes the proof 43 Multiresolution analysis For a given RBF Φ in R n+1 and a sequence of point sets X 1, X 2, S n and a sequence of scales 1, 2, converging to 0, the multiscale approximation scheme in Section 3 can be put into the following multiresolution framework For 1, let W and V be the linear subspaces of H σ (S n ) defined by W := span { φ (, x): x X } and V := span { φ i (, x): x X i, i }, where φ is as in (5) Thus V 1 V 2 V V +1 H σ ( S n) In the language of wavelets, W is the wavelet space and V the scale space The space V can be written as a direct sum of spaces W i, V = W i, 1 1 i and hence if V 0 ={0}, V = V 1 W, 1 That the sum is direct follows under appropriate conditions from the following lemma [1, Lemma 51] Lemma 49 Let Φ be a compactly supported RBF as in [5] and let φ i for i = 1, 2, be scaled SBFs constructed as in (5) where 1, 2, are distinct scales with i 1 LetX i ={x i,1,,x i,ni } S n for i 1 be a set of N i distinct points Then for 1 and a i,k R N i a i,k φ i (, x i,k ) = 0 implies a i,k = 0 for i = 1,, ; k = 1,,N i (29) i=1

12 412 QT Le Gia et al / Appl Comput Harmon Anal 32 (2012) The sequence of spaces {V } is ultimately dense in L 2 (S n ), in the sense of the following theorem [1, Theorem 52] Theorem 410 Let X 1, X 2, be a sequence of point sets on S n with mesh norms h 1, h 2, satisfying cμh h +1 μh for = 1, 2, with fixed μ, c (0, 1) and h 1 sufficiently small Let φ be a kernel satisfying (7) with scale factor = νh and ν satisfying 1/h 1 ν β/μ with a fixed β>0 For all μ sufficiently small ( the closure of V with respect to the norm ) L2 is L 2 S n =1 Acknowledgment The authors are grateful to the Australian Research Council for its support References [1] QT Le Gia, IH Sloan, H Wendland, Multiscale analysis in Sobolev spaces on the sphere, SIAM J Numer Anal 48 (2010) [2] IJ Schoenberg, Positive definite function on spheres, Duke Math J 9 (1942) [3] Y Xu, EW Cheney, Strictly positive definite functions on spheres, Proc Amer Math Soc 116 (1992) [4] C Müller, Spherical Harmonics, Lecture Notes in Math, vol 17, Springer-Verlag, Berlin, 1966 [5] H Wendland, Scattered Data Approximation, Cambridge University Press, Cambridge, 2005 [6] FJ Narcowich, JD Ward, Scattered data interpolation on spheres: error estimates and locally supported basis functions, SIAM J Math Anal 33 (2002) [7] M Abramowitz, I Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, DC, 1970 [8] QT Le Gia, FJ Narcowich, JD Ward, H Wendland, Continuous and discrete least-square approximation by radial basis functions on spheres, J Approx Theory 143 (2006) [9] FJ Narcowich, X Sun, JD Ward, H Wendland, Direct and inverse Sobolev error estimates for scattered data interpolation via spherical basis functions, Found Comput Math 7 (2007) [10] HN Mhaskar, FJ Narcowich, N Sivakumar, JD Ward, Approximation with interpolatory constraints, Proc Amer Math Soc 130 (2002) [11] FJ Narcowich, JD Ward, Scattered-data interpolation on R n : error estimates for radial basis and band-limited functions, SIAM J Math Anal 36 (2004) [12] FJ Narcowich, JD Ward, H Wendland, Sobolev error estimates and a Bernstein inequality for scattered-data interpolation via radial basis functions, Constr Approx 24 (2006)