Asymptotically optimal designs on compact algebraic manifolds

We find t-designs on compact algebraic manifolds with a number of points comparable to the dimension of the space of polynomials of degree t on the manifold. This generalizes results on the sphere by Bondarenko et al. (Ann Math 178(2):443–452, 2013). Of special interest is the particular case of the Grassmannians where our results improve the bounds that had been proved previously.


Introduction
Given a real affine algebraic manifold M endowed with the normalized Lebesgue measure μ M , we say that a collection of N points x 1 , . . . , x N ∈ M is a t-design, also called averaging set or Chebyshev quadrature formula, if As a further evidence of the flexibility of their method we are going to adapt it to the setting of a general algebraic manifold and obtain sharp estimates on the number of points needed for a t-design. The two main ingredients that are needed to adapt their method to this setting are the construction of area regular partitions in manifolds (this has been recently proved by Gigante and Leopardi [18]) and the existence of a sampling-type inequalities (Marcinkiewicz-Zygmund inequalities) for polynomials. This last ingredient essentially follows from a Bernstein-type inequality for polynomials in algebraic varieties proved in [5]. If we specialize our results we confirm the conjecture for Grassmannians stated in [21], and we also find the right order of quadrature formulas on ellipses, as suggested in [23,Remark 3.2].

Definitions and main results
Let M be a smooth, connected and compact affine algebraic manifold of dimension d in R n M = x ∈ R n : p 1 (x) = · · · = p r (x) = 0 , where p 1 , . . . , p r ∈ R[X ] are polynomials with real coefficients and the normal space at x ∈ M is of dimension n − d. This normal space is generated by ∇ p 1 (x), . . . , ∇ p r (x), where ∇ p denotes the gradient of the polynomial p in R n . We consider in M the d−dimensional Hausdorff measure (i.e. the Lebesgue measure) μ M , normalized by μ M (M) = 1. We denote as d(x, y) the geodesic distance between x, y ∈ M.
Let X ⊂ C n be the complexification of M, i.e. the complex zero set of the real polynomials p 1 (x), . . . , p r (x). The Lebesgue measure in X will be denoted by μ X .
The space of real algebraic polynomials on M of total degree at most t, denoted by P t = P t (M) is the restriction to M of the space of real polynomials in n variables. The dimension of the space P t (M) is given by the Hilbert polynomial and it satisfies: Let P 0 t be the Hilbert space of polynomials in P t with zero mean with respect to the usual inner product P 0 t has a reproducing kernel i.e. for each x ∈ M there exists a unique polynomial The existence of designs was already proved in a very general setting in [29], our aim is to show, adapting the techniques from [6] that one can reach the right order given by the following result, see also [12].
Proof Let s be such that 2s = t, if t is even, and such that 2s + 1 = t otherwise. We have that Suppose that N < dim P s . Then for P 1 , . . . , P N ∈ P s linearly independent we have that Indeed, if the determinant above vanishes, there exist a non-trivial linear combination Remark If x 1 , . . . , x N ∈ M is a t-design in M, for even t, and N = dim P t/2 (M), it is easy to see that the set of reproducing kernels of the space P t/2 (M) on those points form an orthogonal basis of P t/2 (M) and for all j = 1, . . . , N . The existence (or not) of these, so-called, tight designs in a variety M seems to be a difficult problem.
In the case of the sphere S d the (sharp) lower bounds tell us that if for t = 2s and t = 2s + 1, respectively. For the sphere S d there are a few tight spherical designs, see [3,4], for which these lower bounds are attained. The tight spherical designs with larger cardinality are the kissing tight 4-design for S 21 of 275 = dim P 2 (S 21 ) points for even t, and the 11−design for S 23 of 196560 points from the Leech lattice for odd t.
Our main result is the following theorem where we show the existence of designs with cardinality N for all N dim P t (M).
Besides the sharp result for the sphere in [6], Kuijlaars has proved on the torus on R 3 the existence of Chebyshev quadratures with Ct 2 points for polynomials of degree t, [24]. In [21] the author obtained results in the very general setting of path-connected topological spaces. His result in our setting provides designs for any N t 2d .
Observe that a sort of converse to Theorem 2.2 also holds. Indeed, let M ⊂ R n be a smooth compact real manifold of dimension d. The existence of a t−design of order t d in M implies that the dimension the space of polynomials in n variables of degree at most t/2, restricted to M, is of order t d . This implies that M is cut out by a system of polynomial equations.
To prove Theorem 2.2, we follow the strategy of [6]. The main ingredients are a result from Brouwer degree theory and Marcinkiewicz-Zygmund inequalities for spaces of polynomials P t (M). In [6] the authors borrow the Marcinkiewicz-Zygmund inequalities on the sphere from [26, Theorem 3.1], see also [11,Theorem 6.4.4]. We will prove the analogue for algebraic polynomials on algebraic varieties.
To state our results we have to define area regular partitions. A finite family of closed sets Following previous constructions for the sphere, it is not difficult to deduce the existence of area regular partitions with diameter comparable to N −1/d for any compact algebraic variety, see for example [28] and the references therein. The existence of such a partition in our case can be deduced also from a recent result by Gigante and Leopardi for Ahlfors regular metric measure spaces, see [18,Theorem 2].

Proposition 2.3 For any N
where B(r ) is a geodesic ball in M of radius r > 0 and the constants c 1 , c 2 depend only on M.
Our result about Marcinkiewicz-Zygmund inequalities is the following: (2). Then for all P ∈ P t for any choice of We will need also Marcinkiewicz-Zygmund inequalities for tangential gradients of polynomials. Observe that, unlike for the sphere, for a general variety the tangential gradient is not necessarily a vector of polynomials in the variety.  (2).
Then for all P ∈ P t As in [6], the last ingredient of the proof of Theorem 2.2 is the following result from Brouwer degree theory: Defining the convenient mapping from P t into itself, this result will give us (1).

Proofs
First we prove the Marcinkiewicz-Zygmund inequalities in the algebraic variety M (Theorem 2.4). Similar results have been obtained also in general compact Riemannian manifolds for spaces of, so-called, diffusion polynomials (i.e. eigenfunctions of elliptic differential operators, in particular, for the Laplace-Beltrami operator), [15,16]. In the proof we use the following result from Berman and Ortega-Cerdà, analogous to the Plancherel-Polya inequality for entire functions of exponential type [30].
Proof of Theorem 2.4 During the proof, we will denote by C any constant depending on M.
Given the degree t ∈ N, let N = t d /a d ≥ At d , where A = A(M) > 0 is some constant to be determined. Consider the area regular partition with N points given by Proposition 2.3. By assumption the balls satisfy and therefore the diameter of R satisfies Then we compare the average of P and the integral and we obtain: where x i is such that |∇ M P(x i )| ≥ |∇ M P(x)| for all x ∈ R i . Observe that we can take x i in the ball B(ac 2 t −1 ) containing R i . Consider now each x i as a point of X , the complex variety, and apply Cauchy's inequality where C is a constant depending on M and B X (x i , t −1 ) is a ball in any hermitian metric in X . Observe that any hermitian metric in X is also a metric when restricted to M and, since M is a compact variety, every metric is comparable. Observe that, for i = 1, . . . , N , the number of balls and R j ⊂ B(x j , Ct −1 ) for j ∈ I. As each R j contains a (disjoint) ball of radius ac 1 t −1 , we get that We can bound the sum on (5) by the sum of integrals on the corresponding balls and using the bounded intersections we can pass to a tubular domain around M defined as in Lemma 3 Finally we apply Lemma 3.1 and we get that (5) is bounded by By using that N = t d /a d and the upper bound for R it is clearly enough to take To prove the Marcinkiewicz-Zygmund inequalities for the tangential gradient (Corollary 2.6) we use the following inequality for vectors of polynomials. (Q 1 (x), . . . , Q m (x)) with m ≤ 2d and Q j (x) ∈ P t+k (M) we have that

t + k holds. That is, for all vectors of polynomials Q(x) =
for any election of x i ∈ R i .
Proof Let A be the constant given by the Theorem 2.4 when we replace t + k with t. Then we use that and we apply the previous result for each Q j (x).
In [6] this result above was enough because the tangential gradient on the sphere of a spherical polynomial can be written as a vector of spherical polynomials (i.e. polynomials restricted to the sphere). This does no longer hold in our case and we have to be more careful.
Proof of Corollary 2.6 Let M be given as the common zero set of the real polynomials Since M is smooth of dimension d, for all x ∈ M the normal space to M on x is generated by where the index i 1 < · · · < i n−d (which may depend on x) is a subset of {1, . . . , r }.
Assume that i j = j for j = 1, . . . , n − d. By the Gram-Schmidt determinant-type formula we obtain an orthogonal basis u 1 (x), . . . , u n−d (x) of the normal space at x by the following determinants Observe that it is a formal determinant that must be computed developing by the last row. Since every ∇ p i (x) is a vector of polynomials, the product ∇ p i (x), ∇ p j (x) is also a polynomial and therefore u i (x) is also a vector of polynomials of total degree bounded by a constant depending only on M. The tangential gradient of P at x ∈ M is then If there are n − d polynomials defining the normal space to M in all the variety, in particular, for the sphere or any other algebraic hypersurface, the result follows because one can apply Corollary 3.2 to the vector of polynomials and use that as M is smooth Clearly, Marcinkiewicz-Zygmund inequalities hold also taking supremum for the subsets I ⊂ {1, . . . , r } with |I | = n − d.
Indeed, now as for some constant C M > 0, the result follows because for v I (x) as in (8) The following result is the main tool to define a mapping with the right properties and apply Theorem 2.7.

Lemma 3.3 There exists a constant A = A(M) > 0 such that if N ≥ At d then there exists a continuous mapping
Proof Let A = A(M) > 0 be given by Corollary 2.6. Let N ≥ At d and R = {R 1 , . . . , R N } be an area regular partition of M as in (2). Given a polynomial P, we define in M the vector field X P = ∇ M P/U (|∇ M P|), where U : R + → R is a smooth increasing function such that is smooth, the vector field X P is smooth on M and depends continuously on P. Now for each 1 ≤ i ≤ N we consider the map y i : [0, ∞) −→ M that satisfies the differential equation where x i ∈ R i . The differential equation changes for each P ∈ P, thus we will sometimes denote y i (s) as y i (P, s) to stress the dependence on P. Note that the quantity N i=1 P(x i ) is small since M P(x)dμ(x) = 0. In order to increase this quantity, we move from the point x i in the direction that increases P(x i ), that is, the direction given by the vector ∇ M P(x i ).
Since the vector field X P is smooth, each y i is well defined and continuous in both P and s. For a fixed s 0 > 0 to be determined, define the continuous mapping P 0 t P → (x 1 (P), . . . , x N (P)) = (y 1 (P, s 0 ) , . . . , y N (P, s 0 )) .
Now, following [6] we split We can modify the area regular partition without losing its essential properties. Indeed, if x i belongs to a ball B(C M R ) containing R i , where C M > 0 is a constant depending only on M then, defining we get an area regular partition with the same properties, i.e.
for some constants c 1 , c 2 depending only on M.
As in (5) and using that P ∈ P 0 t has mean zero, we get where x i is a point in the ball B(c 2 N −1/d ) containing R i where |∇ M P(x)| attains its maximum. Applying the the right-hand side inequality in (4) and the modification of the area regular partition mentioned above we get that For any fixed 0 < s < C M R we apply the left-hand side inequality in (4) and again a modification of the area regular partition to get So, finally taking s 0 = 3K 2 M R and = 1 2K M we get from (11) 1 N N i=1 P(x i (P)) ≥ K M R 2 > 0.

Proof of Theorem 2.2 Fix t and define
= P ∈ P 0 t : which is clearly an open, bounded subset of P 0 t such that 0 ∈ ⊂ P 0 t . Take N ≥ At d for A given by Corollary 2.6 and let x i (P) be the points defined for P ∈ ∂ .
From the continuity of P → (x 1 (P), . . . , x N (P)) it follows that is continuous and from Lemma 3.3, for all P ∈ ∂ , P, P(x i (P)) > 0.
Applying Theorem 2.7 we get the existence of some Q ∈ for which N i=1 K x i (Q) = 0.