Uniformly bounded orthonormal polynomials on the sphere

Given any $\varepsilon>0$, we construct an orthonormal system of $n_k$ uniformly bounded polynomials of degree at most $k$ on the unit sphere in $\mathbb R^{m+1}$ where $n_k$ is bigger than $1-\varepsilon$ times the dimension of the space of polynomials of degree at most $k$. Similarly we construct an orthonormal system of sections of powers $L^k$ of a positive holomorphic line bundle on a compact K\"ahler manifold with cardinality bigger than $1-\varepsilon$ times the dimension of the space of global holomorphic sections to $L^k$.


INTRODUCTION
In [15], the authors construct what are now known as Ryll-Wojtaszczyk polynomials. These are the elements of a sequence {W k } k≥1 of homogeneous polynomials of degree k ≥ 1 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 [1]. For a beautiful monograph about the construction of inner functions in several variables and related problems we refer to [14].
The existence of inner functions in the unit ball B m of C m , for m > 1, was an open problem for many years. As inner functions are (up to multiplication by constants) those f ∈ H 2 (C 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 [3], 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-Wojtaszczyk polynomials, i.e. just 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 [11,Chap. 4]. See also [14, 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-Wojtaszczyk polynomials.
The space of homogeneous holomorphic polynomials of degree k in C m can be identified with the space H 0 (CP m−1 , L k ) of global holomorphic sections of the k-power of the hyperplane bundle L → CP m−1 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 [16], Shiffman constructs a uniformly bounded orthonormal system of sections of powers L k of a positive holomorphic line bundle over a compact Kähler manifold M. He proves that the number n k of sections in the orthonormal system is at least β dim H 0 (M, L k ), where 0 < β < 1 is a number that depends only on the dimension of M. These orthonormal sections are built in [16] by using linear combinations of reproducing kernels peaking at points situated in a latticelike structure on the manifold. In the same paper Shiffman raises the question whether using reproducing kernels peaking at the 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 [13,8]. 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 [16].
One of our main ingredients is that, given ε > 0, it is possible to find interpolating sequences for H 0 (M, L k ) with cardinality (1 − ε) dim H 0 (M, L k ). It is also known, see [8], that there are no (uniform) Riesz basis of reproducing kernels in the space of sections of H 0 (M, L k ). Thus this approach cannot provide uniformly bounded orthonormal basis of sections in H 0 (M, L k ) which would be the ultimate goal.
We will not only consider the complex manifolds setting (as in [16]) but we deal also with a real variant of the problem. In particular we consider spaces generated by eigenfunctions for the Laplace-Beltrami operator in compact two-point homogeneous Riemannian manifolds. The main example is the sphere S m in R m+1 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 + 1 variables restricted to a sphere in R m+1 . 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, [9]. 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.

MAIN RESULTS
Our result in the complex manifold setting reads as: Theorem 2.1. Let L → M be a Hermitian holomorphic line bundle over a compact Kähler manifold M 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: such that s k j ∞ ≤ C ε for 1 ≤ j ≤ n k and for all k ∈ Z + . For M = CP m−1 and L the hyperplane section bundle O(1) endowed with the Fubini-Study metric, one can identify H 0 (CP m−1 , L 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.

Corollary 2.2.
For all m, k ≥ 1 and any ǫ > 0 there is a constant C ε and a system of orthonormal homogeneous holomorphic polynomials In the real setting we consider compact two-point homogeneous Riemannian manifolds. These spaces are essentially the sphere, the projective spaces over the fields R, C and H and the Cayley plane. We introduce the notation. Let (M, g) be a compact two-point homogeneous 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 and we consider the corresponding orthonormal basis of eigenfunctions φ i (so we have ∆φ i = −λ i φ i ). Consider the following subspaces of L 2 (M): We denote dim E L = k L . The reproducing kernels of E L are given by These are functions defined by the properties that, for any w ∈ M, 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 (M ) = B L (w, w) ∼ L m and also k L ∼ L m , see [6]. We denote by b L (z, w) = B L (z, w)/ B L (·, w) L 2 (M ) the normalized reproducing kernels.
Our main example is the sphere M = S m , where the eigenfunctions φ i are spherical harmonics and the spaces E L are the restriction to the sphere of the space of polynomials in R m+1 . In particular, the space of spherical harmonics of degree at most L i.e. the restriction to the sphere S m of the polynomials in m + 1 variables of degree at most L corresponds to E L(L+m−1) .
Our result is the following: Theorem 2.3. Given ε > 0 and L ∈ Z + there exist C ε > 0 and a set {s L 1 , . . . , s L n L } of orthonormal functions in E L with n L ≥ (1 − ε) dim E L such that s L j L ∞ (M ) ≤ C ε , for all L ∈ Z + and 1 ≤ j ≤ n L .
As we mentioned before, we use Fekete arrays in the construction of the orthonormal functions.
is a Fekete set of points of degree L for M.
Fekete points are well suited points for interpolation formulas and numerical integration. One reason is that the corresponding Lagrange polynomials are bounded by 1. 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 M (1) for any {a Lj } L,j with C > 0 independent of L.
It is well known also that the interpolating property is equivalent to say that the Gramian matrix defines a bounded operator in ℓ 2 which is uniformly bounded below, where uniformly means with respect to L, see [4, 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 ∈ M.

Definition 2.6. An array Z = {Z(L)} L≥0 is uniformly separated if there is a positive number
for all L ≥ 0.
The right hand side inequality in (1) holds if and only if Z is uniformly separated, see [9,13]. The next result provides us with Riesz sequences of reproducing kernels with cardinality almost optimal, see [13,8] for a proof in the two different settings we are considering.
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 flat sections. 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.
We consider the following Bochner-Riesz type kernels 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 ∈ M. The main advantage of these modified kernels is that they have better pointwise estimates than the reproducing kernels. The following was proved in [5, Theorem 2.1]: where one can take any N > m (changing the constant). The bound for the reproducing kernel is the same than (2) with N = 1. 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 2.9. Given ε > 0 there exist a set of n L,ε points {z j } j=1,...,n L,ε with n L,ε ≥ (1−ε) dim E L such that the normalized Bochner-Riesz type kernels {b ε L (·, z j )} j=1,...,n L,ε form a Riesz sequence with uniform bounds i.e.
for any {a j } j with C > 0 independent of L.
Proof. We choose the points z j for j = 1, . . . , k (1−2ε)L to be a Fekete array in E (1−2ε)L . It is clear that by an easy application of Theorem 2.7 (enlarging the space instead of shrinking the set of points) they are an interpolating array for E (1−ε)L . We denote n L,ε = k (1−2ε)L .
The right hand side inequality in (3) follows essentially from the uniform separation of the sequence. Indeed, let By duality The set {z j } j=1,...,n L,ε is uniformly separated and Plancherel-Polya inequality says that for all φ ∈ E L , see [12,Theorem 4.6.]. Therefore By Cauchy-Schwarz we get the desired inequality. Given ǫ > 0, let z j for j = 1, . . . , n L,ǫ be the points given by Lemma 2.9 and let be the n L,ε × n L,ε 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) to the Bochner-Riesz type kernel with coefficients given by the function β 2 ǫ (x) getting To define our uniformly bounded functions we will use the entries of matrix ∆ −1/2 . The following localization result by Jaffard, [7,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 2.10 (Jaffard). Let (X, d) be a metric space such that for all ǫ > 0 there exists C ǫ with sup s∈X t∈X exp(−ǫd(s, t)) ≤ C ǫ , and that for a given N > 0 sup s∈X t∈X Let A = (A(s, t)) s,t∈X be a matrix with entries indexed by X and such that for α > N Then, if A is invertible as an operator in ℓ 2 the entries of the matrix A −1 (and also A −1/2 when A is positive definite) satisfies the same kind of bound (5), and therefore the operators defined by these matrices are bounded in ℓ p for 1 ≤ p ≤ ∞, with bounds depending only on the constants C ǫ , 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). As we mentioned before, the reproducing kernel can be bounded with the same bound than in (2) but just with N = 1.  Proof. Let δ > 0 be the separation. Assume that d(z, z j ) ≥ δ/(L + 1) for all j. Then d(z, w) ≤ Using that for geodesic balls V (B(z, r)) ∼ r m one can bound the integral above by a constant times L −m 1 0 t −m/N dt. The case when d(z, z j ) < δ/(L + 1) for some j follows easily. We apply Jaffard's result to the matrix ∆ considering the distance d(i, j) = Ld(z i , z j ) in X = {1, . . . , n L,ǫ } and the estimate (4). The two required properties in Theorem 2.10 can be easily deduced as in Proposition 2.11 so we get To define the orthonormal functions we follow [16]. Denote ∆ −1/2 ij = B ij and define the orthonormal set of functions from E L And then the functions from E L where ζ = e 2πi/n L,ε . They are orthonormal because To verify that the s L i are indeed uniformly bounded we define the linear maps Again, by Proposition 2.11, these maps have ℓ ∞ to L ∞ (M) norm bounded by sup z∈M j |b ε L (z, z j )| L m/2 sup z∈M j 1 (1 + Ld(z, z j )) N L m/2 .
So, finally we get for all L ∈ Z + and 1 ≤ i ≤ n L,ε .