Uniformly bounded sets of orthonormal polynomials on the sphere

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1 Uniformly bounded sets of orthonormal polynomials on the sphere J. Marzo & J. Ortega-Cerdà Universitat de Barcelona October 13, 2014

2 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1.

3 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1. J. Bourgain (85) : There exists a uniformly bounded orthonormal basis in P N = span{z j w N j : j = 0,..., N}, and therefore in H 2 (B 2 ) (closure of A(B 2 ) in L 2 (S 3 )). The construction uses Rudin-Shapiro polynomials.

4 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1. J. Bourgain (85) : There exists a uniformly bounded orthonormal basis in P N = span{z j w N j : j = 0,..., N}, and therefore in H 2 (B 2 ) (closure of A(B 2 ) in L 2 (S 3 )). The construction uses Rudin-Shapiro polynomials. This was an explicit example of RW-sequence (Ryll-Wojtaszczyk).

5 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1. J. Bourgain (85) : There exists a uniformly bounded orthonormal basis in P N = span{z j w N j : j = 0,..., N}, and therefore in H 2 (B 2 ) (closure of A(B 2 ) in L 2 (S 3 )). The construction uses Rudin-Shapiro polynomials. This was an explicit example of RW-sequence (Ryll-Wojtaszczyk). The existence of a RW-sequence (85) implies the existence of a uniformly bounded ON basis in H 2 (B m ).

6 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1. J. Bourgain (85) : There exists a uniformly bounded orthonormal basis in P N = span{z j w N j : j = 0,..., N}, and therefore in H 2 (B 2 ) (closure of A(B 2 ) in L 2 (S 3 )). The construction uses Rudin-Shapiro polynomials. This was an explicit example of RW-sequence (Ryll-Wojtaszczyk). The existence of a RW-sequence (85) implies the existence of a uniformly bounded ON basis in H 2 (B m ). With RW-sequences one can show that inner functions do exist (Aleksandrov s second proof, the first is from 81).

7 Some previous results W. Rudin (80), Inner function Conjecture : There are no Inner functions in B m C m for m > 1. J. Bourgain (85) : There exists a uniformly bounded orthonormal basis in P N = span{z j w N j : j = 0,..., N}, and therefore in H 2 (B 2 ) (closure of A(B 2 ) in L 2 (S 3 )). The construction uses Rudin-Shapiro polynomials. This was an explicit example of RW-sequence (Ryll-Wojtaszczyk). The existence of a RW-sequence (85) implies the existence of a uniformly bounded ON basis in H 2 (B m ). With RW-sequences one can show that inner functions do exist (Aleksandrov s second proof, the first is from 81). It is not known if there exists a uniformly bounded orthonormal basis of holomorphic polynomials in S 2m 1 C m for m 3.

8 Shiffman s result (14) Shiffman constructs a uniformly bounded orthonormal system of sections of powers L N of a positive holomorphic line bundle over a compact Kähler manifold M (i.e. a uniformly bounded orthonormal system of elements of H 0 (M, L N )).

9 Shiffman s result (14) Shiffman constructs a uniformly bounded orthonormal system of sections of powers L N of a positive holomorphic line bundle over a compact Kähler manifold M (i.e. a uniformly bounded orthonormal system of elements of H 0 (M, L N )). The number of sections in the orthonormal system is at least β dim H 0 (M, L N ), for some 0 < β < 1.

10 Shiffman s result (14) Shiffman constructs a uniformly bounded orthonormal system of sections of powers L N of a positive holomorphic line bundle over a compact Kähler manifold M (i.e. a uniformly bounded orthonormal system of elements of H 0 (M, L N )). The number of sections in the orthonormal system is at least β dim H 0 (M, L N ), for some 0 < β < 1. These orthonormal sections are built by using linear combinations of reproducing kernels peaking at points situated in a lattice-like structure on M.

11 Shiffman s result (14) Shiffman constructs a uniformly bounded orthonormal system of sections of powers L N of a positive holomorphic line bundle over a compact Kähler manifold M (i.e. a uniformly bounded orthonormal system of elements of H 0 (M, L N )). The number of sections in the orthonormal system is at least β dim H 0 (M, L N ), for some 0 < β < 1. These orthonormal sections are built by using linear combinations of reproducing kernels peaking at points situated in a lattice-like structure on M. He raises the question whether using kernels peaking at Fekete points one may increase the size of the uniformly bounded orthonormal system of sections.

12 For M = CP m 1 and L the hyperplane section bundle O(1) with the Fubini-Study metric one can identify H 0 (CP m 1, L N ) space of homogeneous holomorphic polynomials of degree N on C m i.e. H 0 (CP 1, L N ) P N. The L p norm of a section is the corresponding norm of the polynomial over the sphere S 2m 1 C m.

13 Theorem Let L 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 > 0 such that for any N Z +, we can find orthonormal holomorphic sections: s N 1,..., s N n N H 0 (M, L N ), n N (1 ε) dim H 0 (M, L N ), such that s N j C ε for 1 j n N and for all N Z +.

14 Let (M, g) be a compact two-point homogeneous Riemannian manifold of dimension m 2. The (discrete) spectrum of the Laplace-Beltrami operator is a sequence of eigenvalues 0 λ 1 λ 2, and we consider the corresponding orthonormal basis of eigenvectors φ i (so we have φ i = λ i φ i ).

15 Let (M, g) be a compact two-point homogeneous Riemannian manifold of dimension m 2. The (discrete) spectrum of the Laplace-Beltrami operator is a sequence of eigenvalues 0 λ 1 λ 2, and we consider the corresponding orthonormal basis of eigenvectors φ i (so we have φ i = λ i φ i ). Consider the following subspaces of L 2 (M): E L = span λi L {φ i }.

16 Let (M, g) be a compact two-point homogeneous Riemannian manifold of dimension m 2. The (discrete) spectrum of the Laplace-Beltrami operator is a sequence of eigenvalues 0 λ 1 λ 2, and we consider the corresponding orthonormal basis of eigenvectors φ i (so we have φ i = λ i φ i ). Consider the following subspaces of L 2 (M): E L = span λi L {φ i }. We denote dim E L = k L. The reproducing kernels of E L are given by k L B L (z, w) = φ i (z)φ i (w). Observe that B L (, w) 2 L 2 (M) = B L(w, w). Hörmander (68) proved that k L B L (w, w) L m. i=1

17 Let (M, g) be a compact two-point homogeneous Riemannian manifold of dimension m 2. The (discrete) spectrum of the Laplace-Beltrami operator is a sequence of eigenvalues 0 λ 1 λ 2, and we consider the corresponding orthonormal basis of eigenvectors φ i (so we have φ i = λ i φ i ). Consider the following subspaces of L 2 (M): E L = span λi L {φ i }. We denote dim E L = k L. The reproducing kernels of E L are given by k L B L (z, w) = φ i (z)φ i (w). Observe that B L (, w) 2 L 2 (M) = B L(w, w). Hörmander (68) proved that k L B L (w, w) L m. We denote by b L (z, w) the normalized reproducing kernels. i=1

18 The main example is the sphere M = S m, where the 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. Our result is the following: Theorem Given ε > 0 and L Z + there exist C ε > 0 and a set {s L 1,..., sl 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.

19 Interpolation and Riesz sequences For degree L we take n L points in M Z(L) = {z L,j M : 1 j n L }, L 0, and assume that n L as L. This yields a triangular array of points Z = {Z(L)} L 0 in M. Definition 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. C 1 n L ˆ a Lj 2 j=1 S d n L a Lj b L (z, z L,j ) j=1 2 n L dσ(z) C a Lj 2, for any {a Lj } L,j with C > 0 independent of L. Observe n L k L. j=1

20 Z is interpolating if and only if the Gramian matrix G = (G ij ) i,j = ( b L (, z L,i ), b L (, z L,j ) ) i,j = (L m/2 b L (z L,i, z L,j )) i,j gives a bounded operator in l 2 which is bounded below (uniformly in L).

21 Z is interpolating if and only if the Gramian matrix G = (G ij ) i,j = ( b L (, z L,i ), b L (, z L,j ) ) i,j = (L m/2 b L (z L,i, z L,j )) i,j gives a bounded operator in l 2 which is bounded below (uniformly in L). The idea is to take G 1/2 = (G 1/2 ij ) ij and for i = 1,..., n L are orthonormal. j G 1/2 ij b L (, z L,j ) E L

22 Z is interpolating if and only if the Gramian matrix G = (G ij ) i,j = ( b L (, z L,i ), b L (, z L,j ) ) i,j = (L m/2 b L (z L,i, z L,j )) i,j gives a bounded operator in l 2 which is bounded below (uniformly in L). The idea is to take G 1/2 = (G 1/2 ij ) ij and j G 1/2 ij b L (, z L,j ) E L for i = 1,..., n L are orthonormal. If one has good estimates for the kernel one can see that for l = 1,..., n L sl L = 1 ζ il nl j j where ζ = e 2πi/n L is bounded and ON. G 1/2 ij b L (, z L,j ) E L

23 Problems to extend the results: It is also known, see (Lev-Ortega-Cerdà (10)) that there are no Riesz basis of reproducing kernels in the space of sections of H 0 (M, L N ). Thus this approach cannot provide uniformly bounded orthonormal basis of sections in H 0 (M, L N ).

24 Problems to extend the results: It is also known, see (Lev-Ortega-Cerdà (10)) that there are no Riesz basis of reproducing kernels in the space of sections of H 0 (M, L N ). Thus this approach cannot provide uniformly bounded orthonormal basis of sections in H 0 (M, L N ). For the sphere S m : Theorem (Bannai, Damerell 79) There is no array Z such that {K L (, z)/ K L (, z) } z Z(L) is an orthonormal basis for the space of spherical harmonics of degree at most L, m 2 and L 3.

25 Problems to extend the results: It is also known, see (Lev-Ortega-Cerdà (10)) that there are no Riesz basis of reproducing kernels in the space of sections of H 0 (M, L N ). Thus this approach cannot provide uniformly bounded orthonormal basis of sections in H 0 (M, L N ). For the sphere S m : Theorem (Bannai, Damerell 79) There is no array Z such that {K L (, z)/ K L (, z) } z Z(L) is an orthonormal basis for the space of spherical harmonics of degree at most L, m 2 and L 3. Open problem It is not known if there are Riesz basis of reproducing kernels in the spaces of spherical harmonics.

26 How to get a maximal Riesz sequence Fekete points (extremal systems) Let {ψ 1,..., ψ kl } be any basis in E L. A set of points x 1,..., x k L M such that det(ψ i (x j )) i,j = max x 1,...,x kl M det(ψ i(x j )) i,j is a Fekete array of points of degree L for M.

27 How to get a maximal Riesz sequence Fekete points (extremal systems) Let {ψ 1,..., ψ kl } be any basis in E L. A set of points x 1,..., x k L M such that det(ψ i (x j )) i,j = max x 1,...,x kl M det(ψ i(x j )) i,j is a Fekete array of points of degree L for M. The next result provides us with Riesz sequences of reproducing kernels with cardinality almost optimal.

28 Theorem (M., Ortega-Cerdà, Pridhnani, Lev) Given ε > 0 let L ɛ = (1 ε)l and Z ε (L) = Z(L ɛ ) = {z Lɛ,1,..., z Lɛ,k Lɛ }, where Z(L) is a set of Fekete points of degree L. Then the array Z ε = {Z ε (L)} L 0 is interpolating i.e. {b L (, z)} z Zε(L) form a Riesz sequence.

29 Theorem (M., Ortega-Cerdà, Pridhnani, Lev) Given ε > 0 let L ɛ = (1 ε)l and Z ε (L) = Z(L ɛ ) = {z Lɛ,1,..., z Lɛ,k Lɛ }, where Z(L) is a set of Fekete points of degree L. Then the array Z ε = {Z ε (L)} L 0 is interpolating i.e. {b L (, z)} z Zε(L) form a Riesz sequence. For compact complex manifolds one can use directly these kernels to continue with the construction of flat sections. In the real setting the off-diagonal decay of the reproducing kernels is not fast enough. So we need to introduce better kernels.

30 Changing the kernel Given 0 < ε 1 let β ε : [0, + ) [0, 1] be a nonincreasing C function such that β ε (x) = 1 for x [0, 1 ε] and β ε (x) = 0 if x > 1. We consider the following Bochner-Riesz type kernels k L BL ε (z, w) = ( ) λi β ε φ k (z)φ k (w). L k=1

31 Changing the kernel Given 0 < ε 1 let β ε : [0, + ) [0, 1] be a nonincreasing C function such that β ε (x) = 1 for x [0, 1 ε] and β ε (x) = 0 if x > 1. We consider the following Bochner-Riesz type kernels k L BL ε (z, w) = ( ) λi β ε φ k (z)φ k (w). L k=1 When ε = 0 we recover the reproducing kernel for E L. Observe that B ε L (, w) 2 2 Lm for any w M. These kernels have better pointwise estimates (Filbir-Mhaskar (10)) B ε L (z, w) L m (1 + Ld(z, w)) N, z, w M where one can take any N > m.

32 One can replace the reproducing kernels by the Bochner-Riesz type and still get a Riesz sequence: Lemma Given ε > 0 there exist a set of n L,ε points {z j } j=1,...,nl,ε with n L,ε (1 ε) dim E L such that the normalized Bochner-Riesz type kernels {bl ε(, z j)} j=1,...,nl,ε form a Riesz sequence (uniformly in L).

33 Jaffard (90) Let (X, d) be a metric space such that for all ɛ > 0 there exists C ɛ such that exp( ɛd(s, t)) C ɛ. Suppose that sup s X t X 1 sup s X 1 + d(s, t) N <, t X and for α > N the matrix A = (A(s, t)) s,t X is such that A(s, t) C 1 + d(s, t) α. Then, if A is invertible as an operator in l 2 the matrix A 1 (and also A 1/2 ) satisfies the same kind of bound and therefore it is bounded in l p for 1 p by Schur s Lemma.

34 We define the n L,ε n L,ε Gramian matrix = ( ij ) i,j=1,...,nl,ε, where ij = b ε L (, z i), b ε L (, z j), where the points z j for j = 1,..., n L,ε are given by the previous Lemma. This matrix defines a bounded operator in l 2 which is also bounded below (uniformly in L). Because of the structure of the regularized kernel we have the following estimate for the entries of the Gramian: ij = 1 k L ˆ M BL ε (z, z i)bl ε(z, z j)dv (z) 1 (1 + Ld(z i, z j )) N.

35 Then as Proposition For {z j } M uniformly separated 1 sup 1. i (1 + Ld(z i, z j )) N j One can apply Jaffard s result getting the estimates 1/2 l l max 1/2 i ij 1. j

36 Denote 1/2 = (B ij ) and define the orthonormal set of functions from E L Ψ L i = B ij bl ε (, z j). j

37 Denote 1/2 = (B ij ) and define the orthonormal set of functions from E L Ψ L i = B ij bl ε (, z j). j And then the polynomials si L = 1 ζ ji Ψ L j, nl,ε where ζ = e 2πi/n L,ε. j

38 Denote 1/2 = (B ij ) and define the orthonormal set of functions from E L Ψ L i = B ij bl ε (, z j). j And then the polynomials si L = 1 ζ ji Ψ L j, nl,ε j where ζ = e 2πi/n L,ε. They are orthonormal because si L, sk L = 1 n L,ε ζ j(i k) = δ ik, 1 i, k n L,ε. n L,ε j=1

39 To verify that the si L are indeed uniformly bounded. Define the linear maps F L : C n L,ε E L, v = (v i ) v j bl ε (, z j). j

40 To verify that the si L are indeed uniformly bounded. Define the linear maps F L : C n L,ε E L, v = (v i ) v j bl ε (, z j). j By the previous Proposition sup z M j b ε L (z, z j) L m/2 sup z M BL ε (z, z j) L m/2. j

41 To verify that the si L are indeed uniformly bounded. Define the linear maps F L : C n L,ε E L, v = (v i ) v j bl ε (, z j). j By the previous Proposition sup z M So, finally we get j b ε L (z, z j) L m/2 sup z M BL ε (z, z j) L m/2. si L L (M) 1 F L l L nl,ε (M) 1/2 l l 1, j for all L Z + and 1 i n L,ε.

Uniformly bounded orthonormal polynomials on the sphere

Submitted exclusively to the London athematical Society doi:0.2/0000/000000 Uniformly bounded orthonormal polynomials on the sphere Jordi arzo and Joaquim Ortega-Cerdà Abstract Given any ε > 0, we construct