Uniformly bounded orthonormal polynomials on the sphere

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1 Submitted exclusively to the London athematical Society doi:0.2/0000/ Uniformly bounded orthonormal polynomials on the sphere Jordi arzo and Joaquim Ortega-Cerdà Abstract Given any ε > 0, we construct an orthonormal system of n k uniformly bounded polynomials of degree at most k on the unit sphere in R m+ where n k is bigger than ε times the dimension of the space of polynomials of degree at most k. Similarly we construct an orthonormal system of sections of powers A k of a positive holomorphic line bundle on a compact Kähler manifold with cardinality bigger than ε times the dimension of the space of global holomorphic sections to A k.. Introduction In [6], the authors construct what are now known as Ryll-Wotaszczyk polynomials. These are the elements of a sequence {W k } k of homogeneous polynomials of degree k in n complex variables such that they are of L 2 -norm one on the unit sphere in C n and uniformly bounded there. These polynomials have proved to be very useful to construct functions with precise growth restrictions and most notably they can be used to construct inner functions in the unit ball in several variables, see []. For a beautiful monograph about the construction of inner functions in several variables and related problems we refer to [5]. The existence of inner functions in the unit ball B m of C m, for m >, was an open problem for many years. As inner functions are (up to multiplication by constants) those f H 2 (B m ) such that f = f 2, it was natural to try to find first uniformly bounded sequences of polynomials. In the unit ball of B 2 C 2, Bourgain, in [4], found a uniformly bounded othonormal basis of H 2 (B 2 ) by constructing a sequence of bounded orthonormal bases of the spaces of holomorphic homogeneous polynomials in B 2. The same question in higher dimensions remains open. Specifically, it is not known if there exist uniformly bounded orthonormal bases for the spaces of holomorphic homogeneous polynomials in B m, for m > 2. We observe that the existence of Ryll-Wotaszczyk polynomials, i.e. ust one bounded polynomial for each degree, implies that H 2 (B m ) do have a uniformly bounded orthonormal basis formed by polynomials for any m > 2, the idea of the construction is due to Olevskii [2, Chap. 4]. See also [5, Appendix I] where a better bound (independent of the dimension) for the elements of the basis is obtained by using powers of an inner function instead of Ryll-Wotaszczyk polynomials. The space of homogeneous holomorphic polynomials of degree k in C m can be identified with the space H 0 (CP m, A k ) of global holomorphic sections of the k-power of the hyperplane bundle A CP m which is endowed with the Fubini-Study metric, so that the L 2 norm of the section is the same as the H 2 (B m ) norm of the polynomial. Thus, it is possible to consider the same problem of existence of bounded orthonormal basis in a more general setting. In [8], Shiffman constructs a uniformly bounded orthonormal system of sections of powers A k 2000 athematics Subect Classification 32A35 and 32L05. The authors are ported by the Generalitat de Catalunya (proect 204 SGR 289) and the Spanish inisterio de Economía y Competividad (proect T C02-0 and proect T P) During the revision of this paper, Bourgain presented the construction of a uniformly bounded basis in the spaces of complex homogenous polynomials on the unit ball of C 3, [5].

2 Page 2 of 9 JORDI ARZO AND JOAQUI ORTEGA-CERDÀ of a positive holomorphic line bundle over a compact Kähler manifold. He proves that the number n k of sections in the orthonormal system is at least β dim H 0 (, A k ), where 0 < β < is a number that depends only on the dimension of. These orthonormal sections are built in [8] by using linear combinations of reproducing kernels peaking at points situated in a lattice-like structure on the manifold. In the same paper Shiffman raises the question whether using reproducing kernels peaking at Fekete points one may increase the size of the uniformly bounded orthonormal system of sections. We provide a positive answer to this question. We proceed as follows: It is known that an arbitrarily small perturbation of an array of Fekete points gives an interpolating sequence, see [4, 9]. Then we use Jaffard s theorem on well localized matrices, together with the interpolation property, to deduce that the inverse of the Gramian matrix defined through the kernels defines a bounded operator in l. Finally we construct the bounded orthonormal sections by following the same arguments as in [8]. One of our main ingredients is that, given ε > 0, it is possible to find interpolating sequences for H 0 (, A k ) with cardinality ( ε) dim H 0 (, A k ). It is also known, see [9], that there are no (uniform) Riesz basis of reproducing kernels in the space of sections of H 0 (, A k ). Thus this approach cannot provide uniformly bounded orthonormal basis of sections in H 0 (, A k ) which would be the ultimate goal. We will not only consider the complex manifolds setting (as in [8]) but we deal also with a real variant of the problem. In particular we consider spaces generated by eigenfunctions for the Laplace-Beltrami operator on certain compact Riemannian manifolds. The main example is the sphere S m in R m+ and the corresponding spaces of polynomials of degree at most k. Our aim is to construct many uniformly bounded orthonormal polynomials of degree at most k in m + variables restricted to a sphere in R m+. We observe that there are no orthonormal basis of reproducing kernels for the space of polynomials of degree at most k, as this would be equivalent to the existence of tight spherical 2k design, [2]. It is not known if there are (uniform) Riesz basis of reproducing kernels, [0]. In any case, as before, our approach cannot provide uniformly bounded orthonormal basis. The proof in this real setting has one extra difficulty when compared to the positively curved holomorphic line bundle setting, because in the real setting the off-diagonal decay of the corresponding reproducing kernel is not fast enough to make the same argument work and some changes are needed. Thus we prefer to present the proof of the more delicate problem, i.e. the Riemannian setting, and we will point out along the way which are the relevant changes to make in the complex setting. 2. ain results Our result in the complex manifold setting reads as: Theorem 2.. Let A be a Hermitian holomorphic line bundle over a compact Kähler manifold with positive curvature. Then for any ε > 0, there is a constant C ε such that for any k Z +, we can find orthonormal holomorphic sections: s k,..., s k n k H 0 (, A k ), n k ( ε) dim H 0 (, A k ), such that s k C ε for n k and for all k Z +. For = CP m and A the hyperplane section bundle O() endowed with the Fubini-Study metric, one can identify H 0 (CP m, A k ) with the space H k (B m ) of homogeneous holomorphic polynomials of degree k on C m. Then the theorem above gives us the following result.

3 BOUNDED ORTHONORAL POLYNOIALS Page 3 of 9 Corollary 2.2. For all m, k and any ε > 0 there is a constant C ε and a system of orthonormal homogeneous holomorphic polynomials p k,..., p k n k H k (B m ), n k ( ε) dim H k (B m ), such that p k L (S 2m ) C ε for n k. Let (, g) be a compact Riemannian manifold of dimension m 2. Let dv be the volume element. The (discrete) spectrum of the Laplace-Beltrami operator is a sequence of eigenvalues 0 = λ 0 < λ 2 λ 2 2, and we consider the corresponding orthonormal basis of eigenfunctions φ i (so we have φ i = λ 2 i φ i). Consider the following subspaces of L 2 (): E L = span λi L {φ i }. We denote dim E L = k L. By Weyl s law k L L m. Definition 2.3. We say that a compact Riemannian manifold is admissible if it satisfies the following product property: there exists a constant C > 0 such that for all 0 < ε < and L : E L E εl E L(+Cε). In [3] this notion was introduced to study Fekete points in compact Riemannian manifolds. There it was proved, that the sphere, the proective spaces over the fields R, C and H are admissible manifolds. oreover the product of two admissible manifolds is admissible. The reproducing kernels of E L are given by B L (z, w) = k L i= φ i (z)φ i (w). These are functions defined by the properties that, for any w, we have that B L (, w) E L and φ, B L (, w) = φ(w), when φ E L. It is known that B L (, w) 2 L 2 () = B L(w, w) L m and also k L L m, see [7]. We denote by b L (z, w) = B L (z, w)/ B L (, w) L2 () the normalized reproducing kernels. The main example is the sphere = S m, where the eigenfunctions φ i are spherical harmonics and the spaces E L are the restriction to the sphere of spaces of polynomials in R m+. In particular, the restriction to S m of polynomials of degree at most L corresponds to E L(L+m ). Our result is the following: Theorem 2.4. Let be an admissible manifold. Given ε > 0, there exists C ε > such that for each L Z + there is a set {s L,..., s L n L } of orthonormal functions in E L with n L ( ε) dim E L such that s L L () C ε, for n L. As we mentioned before, we use Fekete arrays in the construction of the orthonormal functions. Definition 2.5. Let {ψ,..., ψ kl } be any basis in E L. A set of points x,..., x k L such that det(ψ i (x )) i, = max x,...,x kl det(ψ i(x )) i,

4 Page 4 of 9 JORDI ARZO AND JOAQUI ORTEGA-CERDÀ is a Fekete set of points of degree L for. Fekete points are well suited points for interpolation formulas and numerical integration. One reason is that the corresponding Lagrange polynomials are bounded by. We use the interpolating properties of these Fekete points. In the following lines we provide the definition of interpolating arrays and some related concepts. For any degree L we take m L points in Z(L) = {z L, : m L }, L 0, and assume that m L as L. This yields a triangular array of points Z = {Z(L)} L 0 in. Definition 2.6. Let Z = {Z(L)} L 0 be a triangular array with m L k L for all L. We say that Z is interpolating for the spaces E L, if for all arrays {c L, } L 0, ml of values such that m L c L, 2 <, (2.) L 0 k L there exists a sequence of functions Q L E L Q L (z L, ) = c L,, for m L, and for all L. uniformly bounded in L 2 () such that An equivalent definition is the following. The array Z is interpolating if and only if the normalized reproducing kernel of E L at the points Z(L) form a Riesz sequence i.e. 2 m L m L m L C a L 2 a L b L (z, z L, ) dv (z) C a L 2, (2.2) for any {a L } L, with C > 0 independent of L. This explains the factor /k L in Definition 2.6; it corresponds to the normalization B L (z L,, z L, ) k L L m. It is well known also that the interpolating property is equivalent to say that the Gramian matrices G L = G = ( b L (, z L,i ), b L (, z L, ) ) i, = (L m/2 b L (z L,i, z L, )) i, define uniformly bounded operators in l 2 which are uniformly bounded below, where uniformly means with respect to L, see [6, p. 66]. A nice property of interpolating sequences is that they are uniformly separated. We denote by d(u, v) the geodesic distance between u, v. Definition 2.7. An array Z = {Z(L)} L 0 is uniformly separated if there is a positive number ε > 0 such that d(z L,, z L,k ) ε, if k, L + for all L 0. The right hand side inequality in (2.2) holds if and only if Z is uniformly separated, see [0, 4]. The next result provides us with Riesz sequences of reproducing kernels with cardinality almost optimal. See [9] for the definition of separation in the complex setting, and [4, 9] for a proof of the following result in the two different settings we are considering.

5 BOUNDED ORTHONORAL POLYNOIALS Page 5 of 9 Theorem 2.8. Let be an admissible manifold. Given L 0 let Z(L) be a set of Fekete points of degree L for. Then, for any ε > 0, the array {Z(L ε )} L 0 is interpolating for the spaces E L, where L ε = ( ε)l. Remark 2.9. This is the only point in the real setting that one uses the hypothesis that is admissible. In [4] the admissibility is needed to establish the conection between Fekete points and interpolating arrays. The equivalent version of Theorem 2.8 that we need is the following: if Z(L) a set of Fekete points of degree L for then Z(L) is interpolating for the spaces E L + ε where L + ε = ( + ε)l. The theorem above shows that the normalized reproducing kernels {b L (, z)} z Z(Lε) form a Riesz sequence. In the compact complex manifolds setting one can use directly these kernels to continue with the construction of sections satisfying the conclusions of Theorem 2.. This is because the Bergman kernel B k of the k th -power of a positively curved holomorphic line bundle over a compact complex manifold of dimension m has very fast off-diagonal decay, see [3]: B k (z, w) k m e c kd(z,w). Unfortunately, when working with Riemannian manifolds the off-diagonal decay of the reproducing kernels is not fast enough. So we are going to introduce better kernels. Definition 2.0. Given 0 < ε let β ε : [0, + ) [0, ] be a nonincreasing C function such that β ε (x) = for x [0, ε] and β ε (x) = 0 if x >. We consider the following Bochner-Riesz type kernels k L ( ) BL(z, ε λk w) = β ε φ k (z)φ k (w). L k= In the limiting case, when ε = 0, we recover the reproducing kernel for E L. Observe that one obtains easily, from the corresponding result for the reproducing kernel, that B ε L (, w) 2 2 L m for any w. The main advantage of these modified kernels is that they have better pointwise estimates than the reproducing kernels. The following was proved in [7]: BL(z, ε L m w) C ε,n, z, w (2.3) ( + Ld(z, w)) N where one can take any N > m (changing the constant). The bound for the reproducing kernel is the same than (2.3) with N =. As before we denote by lower-case b ε L (z, w) the normalized kernel. Our next result shows that one may replace the reproducing kernels by the Bochner-Riesz type and still get a Riesz sequence. Lemma 2.. Given ε > 0 there exist a set of points {z },...,nl,ε with ( ε) dim E L such that the normalized Bochner-Riesz type kernels {b ε L (, z )},...,nl,ε form a Riesz sequence with uniform bounds i.e. C a 2 a b ε L(z, z ) for any {a } with C > 0 independent of L. 2 n L,ε dv (z) C a 2, (2.4)

6 Page 6 of 9 JORDI ARZO AND JOAQUI ORTEGA-CERDÀ Proof. We choose the points z for =,..., k ( 2ε)L to be a Fekete array in E ( 2ε)L. It is clear that by an application of Theorem 2.8 (see Remark 2.9) they are an interpolating array for E ( ε)l. We denote = k ( 2ε)L. The right hand side inequality in (2.4) follows essentially from the uniform separation of the sequence. Indeed, let n L,ε S L,ε (a) = a b ε L(z, z ). By duality and B ε L (, w) 2 2 L m : S L,ε (a) 2 = P, S L,ε (a) P 2= L m/2 P 2= n L,ε a BL(z ε, w)p (w)dv (w). The set {z },...,nl,ε is uniformly separated and Plancherel-Polya inequality says that φ(z ) 2 φ 2 2, k L for all φ E L, see [3, Theorem 4.6.]. Therefore BL(z ε, w)p (w)dv (w) 2 k L BL(, ε w)p (w)dv (w) 2 P By Cauchy-Schwarz we get the desired inequality. For the left hand side in (2.4), let P ( ε)l the orthogonal proection from L 2 (S d ) onto E ( ε)l. Then as P ( ε)l (BL ε (, w))(z) is the reproducing kernel in E ( ε)l, and z for =,..., is interpolating for E ( ε)l a 2 P[( ε)l] S L,ε (a) 2 S 2 L,ε(a) 2 2. Given ε > 0, let z for =,..., be the points given by Lemma 2. and let b ε L (, z ), b ε L (, z ) b ε L (, z ), b ε L (, z ) = ( i ) i,,...,nl,ε =.. b ε L (, z ), b ε L (, z ) b ε L (, z ), b ε L (, z ) be the corresponding Gramian matrix. This matrix defines a uniformly bounded operator in l 2 which is also bounded below uniformly. It is clear that one can apply the estimate (2.3) to the Bochner-Riesz type kernel with coefficients given by the function βε(x) 2 getting i L m A BL(z, ε z i )BL ε (z, z )dv (z) L m dv (z) ( + Ld(z, z i )) N ( + Ld(z, z )) N. We are going to estimate the last integral. Define the sets A = {z : 2d(z, z i ) d(z i, z )} and B = {z : 2d(z, z ) d(z i, z )}. Clearly = A B, and L m dv (z) ( + Ld(z, z i )) N ( + Ld(z, z )) N L m dv (z) ( + Ld(z i, z )) N ( + Ld(z, z )) N. Now dv (z) = Lm ( + Ld(z, z)) N 0 V ({ w : ( + Ld(z, w)) N > t }) dt, (2.5)

7 BOUNDED ORTHONORAL POLYNOIALS Page 7 of 9 and using that for geodesic balls V (B(z, r)) r m one can bound the integral above by a constant times L m 0 t m/n dt. The same estimate holds for the integral over B, therefore we have proved that i ( + Ld(z i, z )) N. (2.6) To define our uniformly bounded functions we will use the entries of matrix /2. The following localization result by Jaffard, [8, Proposition 3], say basically that if we have an invertible matrix in l 2 which is well localized on the diagonal, its inverse matrix is also localized along the diagonal and thus bounded in l p. Theorem 2.2 (Jaffard). C ε with and that for a given N > 0 Let (X, d) be a metric space such that for all ε > 0 there exists exp( εd(s, t)) C ε s X t X s X t X ( + d(s, t)) N = B <. Let A = (A(s, t)) s,t X be a matrix with entries indexed by X and such that for α > N A(s, t) C ( + d(s, t)) α. (2.7) Then, if A is invertible as an operator in l 2 the entries of the matrix A (and also A /2 when A is positive definite) satisfies the same kind of bound (2.7), and therefore the operators defined by these matrices are bounded in l p for p, with bounds depending only on the constants C ε, B and C. The following proposition allow us to apply Jaffard s result to the matrix and it will be used also to bound our orthonormal functions. We observe that it is precisely in this point where we need estimate (2.3). As we mentioned before, the reproducing kernel can be bounded with the same bound as in (2.3) but ust with N =. Proposition 2.3. Let {z } be uniformly separated and let N > m. Then. ( + Ld(z, z )) N z Proof. Let δ > 0 be the separation. Assume that d(z, z ) δ/(l + ) for all. Then d(z, w) 3 2 d(z, z ) for w B = {w : d(z, w) < δ/2(l + )}. Therefore, arguing as in (2.5) we get (+Ld(z,z )) L m N B dv (w) (+Ld(z,w)) N L m dv (w) C. (+Ld(z,w)) N The case when d(z, z ) < δ/(l + ) for some follows easily. We apply Jaffard s result to the matrix considering the distance d(i, ) = Ld(z i, z ) in X = {,..., } and the estimate (2.6). The two required properties in Theorem 2.2 can be

8 Page 8 of 9 JORDI ARZO AND JOAQUI ORTEGA-CERDÀ easily deduced as in Proposition 2.3 so we get /2 i ( + Ld(z i, z )) N, and /2 l l = max i /2 i. To define the orthonormal functions we follow [8]. Denote /2 i = B i and define the orthonormal set of functions from E L Ψ L i = B i b ε L(, z ). And then the functions from E L s L i = ζ i Ψ L, nl,ε where ζ = e 2πi/. They are orthonormal because To verify that the s L i s L i, s L k = n L,ε ζ (i k) = δ ik, i, k. are indeed uniformly bounded we define the linear maps F L : C E L, v = (v i ) v b ε L(, z ). Again, by Proposition 2.3, these maps have l to L () norm bounded by b ε L(z, z ) L m/2 z z ( + Ld(z, z )) N Lm/2. So, finally we get for all L Z + and i. s L i L () F L l L () /2 l l, nl,ε Acknowledgements. We would like to thank the referee for a careful review of the paper. References. A. Alexandrov, Inner functions on compact spaces, Functional. Anal. Appl. 8 (984), pp E. Bannai, R.. Damerell, Tight spherical designs, I. J. ath. Soc. Japan 3, no., pp , B. Berndtsson, Bergman kernels related to Hermitian line bundles over compact complex manifolds, Explorations in complex and Riemannian geometry, Contemp. ath., vol. 332, Amer. ath. Soc., Providence, RI, 2003, pp J. Bourgain, Applications of the spaces of homogeneous polynomials to some problems on the ball algebra, Proc. Amer. ath. Soc. 93 (985), no. 2, pp J. Bourgain, On uniformly bounded basis in spaces of holomorphic functions, (205), arxiv: v [math.fa]. 6. O. Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis. Birkhuser Boston, Inc., Boston, A, L. Hörmander, The spectral function of an elliptic operator, Acta ath. 2, no. (968), pp S. Jaffard, Propriétés des matrices bien localiséees près de leur diagonale et quelques applications. (French) [Properties of matrices well localized near their diagonal and some applications] Ann. Inst. H. Poincar Anal. Non Linaire 7 (990), no. 5, pp

9 BOUNDED ORTHONORAL POLYNOIALS Page 9 of 9 9. N. Lev, J. Ortega-Cerdà, Equidistribution estimates for Fekete points on complex manifolds, J. Eur. ath. Soc., to appear. (203) arxiv: [math.cv]. 2, 4 0. J. arzo, arcinkiewicz-zygmund inequalities and interpolation by spherical harmonics, J. Funct. Anal. 250 (2007), no. 2, pp , 4. J. arzo, J. Ortega-Cerdà, Equidistribution of Fekete points on the sphere, Constr. Approx. 32 (200), no. 3, pp A. Olevskii, Fourier series with respect to general orthogonal systems, Ergebnisse der athematik und ihrer Grenzgebiete, Band 86. Springer-Verlag, New York-Heidelberg, J. Ortega-Cerdà, B. Pridhnani, Carleson measures and Logvinenko-Sereda sets on compact manifolds, Forum ath. 25 (20),, pp , 6 4. J. Ortega-Cerdà, B. Pridhnani, Beurling-Landau s density on compact manifolds, J. Funct. Anal. 263 (202) pp , 4, 5 5. W. Rudin, New constructions of functions holomorphic in the unit ball of C n, CBS Regional Conference Series in athematics, 63, AS, Providence, RI, J. Ryll, P. Wotaszczyk, On homogeneous polynomials on a complex ball, Trans. Amer. ath. Soc. 276 (983), pp C.D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of ath. 26 (987), pp B. Shiffman, Uniformly bounded orthonormal sections of positive line bundles on complex manifolds Proceedings of the Conference on Analysis, Complex Geometry, and athematical Physics, Contemporary athematics, Amer. ath. Soc., Providence, RI (to appear)., arxiv: [math.cv]., 2, 8 J. arzo and J. Ortega-Cerdà Dpt. de atemàtica Aplicada i Anàlisi Universitat de Barcelona. Gran Via 585, Barcelona. Spain marzo@ub.edu ortega@ub.edu