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, Radchenko and Viazovska. 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 for all polynomials P (x) of total degree less or equal than t.
The results for Chebyshev quadrature on the interval are classic, see for example the survey paper by Gautschi and the commentaries by Korevaar, [Gau14].
In the case of the sphere, these sets are known as spherical designs and were introduced by Delsarte, Goethals and Seidel [DGS77] in the context of algebraic combinatorics on spheres.
Since then, spherical designs have gained popularity in different areas of mathematics, ranging over geometry, algebraic and geometric combinatorics, and numerical analysis. See for instance the review [BB09] by Bannai and Bannai or the more recent review of Brauchart and Grabner [BG15].
Very early the notion of t-designs was considered in other contexts beyond the sphere. The projective designs were introduced by Hoggard and further investigated by Lyubich and Shalatova, [LS04], as a tool to embed isometrically ℓ 2 (R n ) in ℓ 2t (R m ).
When they are separated, the points in t-designs tend to be evenly distributed along the sphere or the projective space, and they are close to optimal for other problems, like minimization of Newtonian energy as observed in [HL08] and [BRV15].
For this reason, in order to have evenly distributed subspaces, the tdesigns on Grassmannians were considered by Bachoc, Coulageon and Nebe Date: December 21, 2016. This research has been partially supported by MTM2014-57590-P and MTM2014-51834-P grants by the Ministerio de Economía y Competitividad, Gobierno de España and by the Generalitat de Catalunya (project 2014 SGR 289).
in [BCN02]. In [BEG16] Breger, Ehler and Gräf make some numerical study of approximation problems in Grassmannians and make the observation that two possible notions of designs in Grassmannians (one of them using polynomials as we did and another when one replaces polynomials by eigenvectors of the Laplacian as in [BCN02]) are indeed related.
The existence of designs in a very general setting was proved by Seymour and Zaslavsky in [SZ84], see also [AdR88]. One challenging problem is to find designs with as few points as possible for every t.
In [Kan15] Kane studied the existence of designs in path-connected spaces. In particular for the Grasmmanian G(k, R n ), linear subspaces of R n of dimension k, he proved that one can find t-designs with a number of points of order O(t 2k(n−k) ) and he conjectures that it should be of order O(t k(n−k) ) since this is the dimension of the corresponding space of polynomials.
A breakthrough on the existence of designs of size of optimal asymptotic order was obtained in [BRV13], where the conjecture by Korevaar and Meyers was settled. There are t-designs with O(t d ) points in the d-dimensional sphere. This is asymptotically the best rate possible. The proof is obtained by a fixed point theorem, thus it is not constructive. On the other hand their method is very flexible, so they could improve it in [BRV15] to obtain separated spherical designs with the same cardinality.
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 in [GL15]) and the existence of a samplingtype inequalities (Marcinkiewicz-Zygmund inequalities) for polynomials for sequences of points sufficiently close and separated. This last ingredient essentially follows from a Bernstein-type inequality for polynomials in algebraic varieties proved in [BOC15]. If we specialize our results to the Grassmannians we confirm the conjecture in [Kan15].

Definitions and Main results
Let M be a smooth, connected and compact affine algebraic manifold of where p 1 , . . . , p r ∈ R[X] are polynomials with real coefficients and the normal space at x ∈ M, generated by ∇p 1 (x), . . . , ∇p r (x), is of dimension n−d. 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 this space has a reproducing kernel i.e. for each x ∈ M there exists a unique polynomial K x ∈ P 0 t such that The existence of designs was already proved in a very general setting in [SZ84], our aim is to show, adapting the techniques from [BRV13], that one can reach the right order, given by the following result.
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 det(P j (x i )) i,j=1,...,N = 0. Indeed, if the determinant above vanishes, there exist a non-trivial linear combination and we get a contradiction from 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 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 [BD79,BD80], for which these lower bound 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 [BRV13], Kuijlaars has proved on the torus on R 3 the existence of Chebyshev quadratures with Ct 2 points for polynomials of degree t, [Kui95]. In [Kan15] 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 so twice as much as in our result.
To prove Theorem 2.2, we follow the strategy of [BRV13]. The main ingredients are a result from Brouwer degree theory and Marcinkiewicz-Zygmund inequalities for spaces of polynomials P t (M). In [BRV13] the authors borrow the Marcinkiewicz-Zygmund inequalities on the sphere from [MNW01, Theorem 3.1], see also [DX13, 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 R 1 , . . . , R N ⊂ M is an area regular partition of M if The diameter of the partition R = {R 1 , . . . , R N } is 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 [RSZ94] and the references therein. The existence of such a partition in our case can be deduced also from a recent result by Gigante 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 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 polynomial.  Then for all P ∈ P t (4) where K M = 3 √ d r n−d C M for C M > 0 depending only on M and for any choice of x i ∈ R i , with 1 ≤ i ≤ N .
As in [BRV13], the last ingredient of the proof of Theorem 2.2 is the following result from Brouwer degree theory: Theorem 1.2.9]). Let f : R n −→ R n be a continuous mapping and Ω an open bounded subset, with boundary ∂Ω, such that 0 ∈ Ω ⊂ R n . If x, f (x) > 0 for all x ∈ ∂Ω, then there exists x ∈ Ω satisfying f (x) = 0.
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), [FM10,FM11]. In the proof we use the following result from Berman and Ortega-Cerdà [BOC15, Theorem 10]. This result is analogous to Plancherel-Polya inequality for entire functions of exponential type, [You01].
Lemma 3.1. There exists a constant C = C M > 0 such that for all polynomials P ∈ P t the following inequality holds where C = C(M) > 0 is a constant depending only M and 1 ≥ aA. Just take A ≥ 2C. During all the proof, we will call C all the constants depending on M. Let N = t d /a d , for some constant a > 0 such that 1 ≥ aA. By assumption 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 X. We assume that a < 1/2c 2 and then B We can bound the sum on the balls by the integral on a tubular domain around M defined as in Lemma 3.1 (see [BOC15]) and taking into account the multiplicities: Finally, we apply Lemma 3.1 and we get that (5) is bounded by so by using that N = t d /a d and the upper bound for R we get the result.
To prove the Marcinkiewicz-Zygmund inequalities for the tangential gradient (Corollary 2.6) we use the following inequality for vectors of polynomials.
Corollary 3.2. Let k ∈ N be a fixed constant. There exists a constant A = A(M, k) > 0 such that if N ≥ At d and R = {R 1 , . . . , R N } is an area regular partition as in (2). Then, for all vector of polynomials Q(x) = (Q 1 (x), . . . , Q m (x)) with m ≤ 2d and Q j (x) ∈ P t+k (M) we have that for any election of x i ∈ R i .
Proof. Let A be the constant given by the previous theorem when we replace t + k instead of t. Then we use that and we apply the previous result for each Q j (x).
In [BRV13] 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). In our case this is no longer the 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 p 1 (x), . . . , p r (x).
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 determinanttype 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 since every ∇p i (x) is a vector of polynomials, the product ∇p i (x), ∇p j (x) is also a polynomial an 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 for some C M > 0. Now for any I ⊂ {1, . . . , r} with |I| = n − d we can define the vectors of polynomials u I j (x) for j = 1, . . . , n − d (where maybe some of the polynomials are zero) and by the previous corollary the Marcinkiewicz-Zygmund inequalities hold for 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)  such that for all P ∈ P 0 t with M |∇ t P (x)|dµ(x) = 1 P, 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 = ∇ t P/U ǫ (|∇ t P |), where U ǫ : R + → R is a smooth increasing function such that U ǫ (x) = ǫ/2 if 0 ≤ x ≤ ε/2 and U ǫ (x) = x if x ≥ ε for some ǫ fixed. Since U ǫ (x) 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 (9) ∂ ∂s y i (s) = X P (y i (s)) y i (0) = x i 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 .
Remark. 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 ∇ t 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 (10) P 0 t ∋ P → (x 1 (P ), . . . , x N (P )) = (y 1 (P, s 0 ) , . . . , y N (P, s 0 )) . Now, following [BRV13] we split Observe that 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. B(c ′ 1 N −1/d ) ⊂ R ′ i ⊂ B(c ′ 2 N −1/d ), 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 |∇ t P (x)| attains its maximum. Applying Corollary 2.6 we get that For any fixed 0 < s < C M R we apply the remark above about the area regular partition and Corollary 2.6 to get d ds |∇ t P (y i (P, s))| 2 U ǫ (|∇ t P (y i (P, s))|) ≥ 1 N i : |∇tP (y i (P,s))|≥ǫ |∇ t P (y i (P, s))| ≥ 1 N N i=1 |∇ t P (y i (P, s))| − ǫ So, finally taking s 0 = 3K 2 M R and ǫ = 1 2K M we get from (11) Proof of Theorem 2.2. Fix t and define Ω = P ∈ P 0 t : which is clearly an open, bounded subset of P 0 t such that 0 ∈ Ω ⊂ P 0 t . Take N ≥ At d for A given by the previous lemma and let x i (P ) be the points defined for P ∈ ∂Ω.
Applying Theorem 2.7 we get the existence of some Q ∈ Ω for which N i=1 K x i (Q) = 0.