Reproducing Kernels for the Irreducible Components of Polynomial Spaces on Unions of Grassmannians

The decomposition of polynomial spaces on unions of Grassmannians Gk1,d∪…∪Gkr,d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {G}}_{{k_1},d}\cup \ldots \cup \mathcal G_{{k_r},d}$$\end{document} into irreducible orthogonally invariant subspaces and their reproducing kernels are investigated. We also generalize the concepts of cubature points and t-designs from single Grassmannians to unions. We derive their characterization as minimizers of a suitable energy potential to enable t-design constructions by numerical optimization. We also present new analytic families of t-designs for t=1,2,3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1,2,3$$\end{document}.

a finite sum over the sampling values. However, many open questions remain when dealing with polynomials on unions of nonconnected manifolds.
Orthogonal projectors with fixed rank are used in many applications for analysis and dimension reduction purposes, cf. [31,44], leading to a function approximation problem on a single Grassmannian manifold. Projections with varying target dimensions are more flexible and may offer a larger range of applications. Therefore, we shall study unions of Grassmannians.
By studying the structure of polynomial spaces on the union of Grassmannians, some of our findings generalize results in [34]. In particular, we shall verify that the multiplicities of the irreducible representations of the orthogonal group occurring in an orthogonally invariant reproducing kernel Hilbert space on unions of Grassmannians coincide with the ranks of the kernel's Fourier coefficients. This enables us to actually determine the multiplicities in the space of polynomials of degree t. Moreover, we construct the underlying reproducing kernels for the irreducible components. While cubatures and t-designs in single Grassmannians have been studied in [2][3][4][5], we shall also investigate these concepts in unions of Grassmannians. We derive a characterization as minimizers of an energy functional induced by a reproducing kernel. By numerically minimizing the energy functional, we compute candidates for t-designs, i.e., t-designs up to machine precision. We are then able to describe these candidates analytically and check that the energy functional is minimized exactly.
It should be mentioned that the topic shares some common themes with Euclidean designs, cf. [40], where unions of spheres with varying radii in Euclidean space are considered, see also [7][8][9][10]. The structure of the polynomial spaces on unions of spheres have been investigated in [21], but the ideas in those proofs do not work for unions of Grassmannians, whose structure appears to be more involved.
The outline is as follows. In Sect. 2 we recall some facts on polynomial spaces on single Grassmannians and their irreducible decompositions. Section 3 is dedicated to some elementary results on polynomial spaces on unions of Grassmannians. Direct consequences of irreducible decompositions of polynomials on symmetric matrices are studied in Sect. 4. In Sect. 5 we determine the multiplicities of the polynomial spaces on unions and construct the underlying reproducing kernels for the irreducible components. In Sect. 6 we introduce cubatures and t-designs on unions of Grassmannians and derive a characterization as minimizers of an energy functional induced by a reproducing kernel. We compute some analytical minimizers in Sect. 7.

Polynomials on Single Grassmannians
This section is dedicated to summarizing some facts about single Grassmannians, see, for instance, [5,34]. The Grassmannian space of all k-dimensional linear subspaces of R d is naturally identified with the set of orthogonal projectors on R d of rank k denoted by G k,d := {P ∈ R d×d sym : P 2 = P; Tr(P) = k}, where R d×d sym is the set of symmetric matrices in R d×d . Each Grassmannian G k,d admits a unique orthogonally invariant probability measure σ k,d induced by the Haar (probability) measure σ O(d) on the orthogonal group O(d); i.e., for any Q ∈ G k,d and measurable function f , The space of complex-valued, square-integrable functions L 2 (G k,d ), endowed with the inner product ( f , g) G k,d , decomposes into orthogonally invariant subspaces where H λ (G k,d ) is equivalent to H d 2λ , the irreducible representation of O(d) associated with the partition 2λ = (2λ 1 , . . . , 2λ t ), cf. [5,34]. Note that two representations are equivalent if there is a linear isomorphism that commutes with the group action. A partition of t is an integer vector λ = (λ 1 , . . . , λ t ) with λ 1 ≥ · · · ≥ λ t ≥ 0, |λ| = t, where |λ| := An orthogonally invariant measure on G I,d is derived by the sum of the corresponding measures on the single Grassmannians. In Sect. 6, we shall also allow for weighted sums. According to (1), the corresponding space of complex-valued, square-integrable functions L 2 (G I,d ) decomposes into where d I is the set of all partitions λ of length at most max k∈I (min(k, d − k)) and the multiplicities are the cardinality of As for a single Grassmannian, we consider polynomials on G I,d given by multivariate polynomials in the matrix entries of a given projector P ∈ G I,d , i.e., This space decomposes orthogonally into where the multiplicities μ d λ (I, t) still need to be determined. Indeed, this is the topic of the first part of the present paper.
Proof The cases t = 0 and |I| = 1 are trivially fulfilled. Suppose t ≥ 1 and r ≥ 2. The restriction mapping An induction over t and r completes the proof.
Reformulation yields μ d λ (I, |λ|+s) ≥ min{s +1, |I d λ |} for s ≥ 0. Due to (2), the upper bound μ d λ (I, t) ≤ |I d λ | holds. Subsequent sections shall reveal that equality holds in (3). However, this requires a closer look at relations among irreducible representations and their reproducing kernels; see Appendix A.1 for some basics on reproducing kernels that shall be used in the following.

Determining the Multiplicities for a Few Special Cases
The space of polynomials on R d×d sym of degree at most t and its subspace of homogeneous polynomials of degree t are denoted by respectively. The differential inner product between f , g ∈ Pol t (R d×d sym ) given by is orthogonally invariant, cf. [43], inducing the orthogonal decomposition Hom s (R d×d sym ).

Remark 4.1
In accordance with [30], the mapping (X , Y ) → 1 s! Tr(XY ) s is the reproducing kernel for Hom s (R d×d sym ) with respect to the differentiation inner product, cf. also Appendix A.1.

Remark 4.2
One can check that the function (X , Y ) → 1 |λ|! C λ (XY ) is the reproducing kernel for F λ (R d×d sym ) with respect to the differentiation inner product, where C λ is the zonal polynomial of index λ, cf. [30] and Appendix B.1.
By restricting the group action from GL(d) to O(d), the space F λ (R d×d sym ) decomposes further as where the multiplicities ν d 2λ,λ ∈ N 0 are determined by the corresponding branching rule, cf. [26,33,35]. If λ is such that H d λ is not defined, then we simply put ν d 2λ,λ = 0. We note that Thus, we obtain μ d λ (I, t) = 1, for |λ| = t ≥ 0 with λ ∈ d I . The latter enables us to verify that the lower bounds in Corollary 3.2 are an equality for |I| = 2. Indeed, if we order which means that equality holds in (3). For arbitrary I ⊂ {1, . . . , d − 1}, we can still use (5) to determine the multiplicities μ d λ (I, t) provided that t is sufficiently small, but this takes some preparation. The space of homogeneous polynomials restricted to G I,d is denoted by Hom t (G I,d ) : It is important to notice that the restriction of Hom t (R d×d sym ) to G I,d yields (almost) the entire space Pol t (G I,d ): Proof First we note that Tr(X ·) t , X ∈ R d×d sym , linearly generates the space Hom t (G I,d ), cf. Remark 4.1, (27), and (28). Now, for t ≥ 1 and I ⊂ {1, . . . , d − 1}, we have Tr(X P) s = Tr(X P) s−1 Tr(X P t−s+1 ), Since the term on the right-hand side is a homogeneous polynomial of degree t in P restricted to G I,d , we deduce . According to invariant theory, the ring of orthogonally invariant polynomials on R d×d sym is generated by polynomials of the form Tr(X l ), X ∈ R d×d sym , l ∈ N 0 , cf. [42,Theorem 7.1]. Since f • is also homogeneous of degree t,  For any constant C > 0, the map (P, Q) → Tr(P Q) t + C is a reproducing kernel for Pol t (G I,d ).
Next, we determine the multiplicities μ d λ (I, t), for t = 1, 2, 3, by deriving upper bounds that match the lower bounds in Corollary 3.2. The decompositions (4) and (5) yield and The multiplicities of H d λ in Hom t (R d×d sym ) are upper bounds for μ d λ (I, t) since the restriction mapping is orthogonally invariant. For t = 1, 2, the lower bounds in Corollary 3.2 are matched, so that we have determined μ d λ (I, 1) and μ d λ (I, 2) for any index , the analysis is more difficult, and we observe that the branching rules yield The multiplicity of H d (2) in Hom 3 (R d×d sym ) does not match the lower bound in Corollary 3.2. Instead, we found that the kernel which may not have been observed in the literature yet, reproduces a subspace of 2) , for d > 2, and equivalent to H d (2) , for d = 2, respectively. Since K vanishes on any Grassmannian, i.e., K (X , Y ) = 0, for all Y ∈ G k,d , k ∈ {1, . . . , d − 1} and X ∈ R d×d sym , we deduce that the multiplicity of H d (2) in Pol 3 (G I,d ) is less than in Hom 3 (R d×d sym ). Now, for even partitions, the resulting upper bounds on the multiplicities match the lower bounds in Corollary 3.2 for t = 3 and any index set To determine the multiplicities of the irreducible subspaces of Pol 3 (G I,d ) for |I| ≥ 3, we used the reproducing kernel K in (8). We shall further explore the reproducing kernels of the irreducible components to determine the multiplicities in Pol t (G I,d ) for |I| ≥ 3 and t ≥ 4.

Zonal Kernels and Harmonic Analysis on Unions of Grassmannians
In particular, T k,l λ commutes with complex conjugation and the group action. It can be realized by an integral transform with a unique real-valued zonal function p k,l λ : Note that zonal means p k,l λ were studied in [34] and expanded into the zonal polynomials by λ,λ = 1, and q λ,λ is a polynomial of degree |λ| − |λ | given by Potential zeros in the denominator of (x) λ (x) λ cancel out, so that the fraction is well defined. Up to the scaling, which we have not specified yet, the functions p k,k λ are the reproducing kernels for H λ (G k,d ) with respect to the L 2 inner product when 1 ≤ k ≤ d/2 and (λ) ≤ k, cf. [34].
The sum of the right-hand side in (10) . One of our contributions going beyond [34] is to determine the reproducing kernel of H |λ| λ (G d ) with the help of a particular extension of (10) to this broader range of parameters. Recall that In order to verify Theorem 5.1, we choose a suitable normalization of the intertwining functions, induced by the following selection of intertwining operators.
such that the following diagram commutes: and straightforward calculations yield the statement.
The integral operators in Proposition 5.2 induce intertwining functions p k,l λ via (9) satisfying where the Fourier coefficientsf k,l λ and the basis functions P k,l λ are arranged in matrix formf Convolving two continuous zonal functions yields again a continuous zonal function f * g. It is straightforward to check that its 4 This convolution property implies that the kernel (P, Q) → Tr(P λ (P, Q)), are positive semidefinite matrices and thus allow for a spectral decomposition where λ | are orthogonal rank-1 projectors corresponding to an eigenbasis ofK λ . The corresponding kernels are also positive definite. This spectral decomposition of K yields the irreducible decomposition of the underlying reproducing kernel Hilbert space denoted by S(K ), see also (27):  (14) and (15) Proof Mercer's theorem implies that the reproducing kernel Hilbert space S(K ) decomposes into the pairwise orthogonal eigenspaces with nonzero eigenvalues associated with the integral operator T K : This decomposition corresponds to the eigenspace decomposition of T K in the subspace H λ (G I,d ). More precisely, the convolution property yields that the kernels K i λ satisfy, for i, j = 1, . . . , |I d λ |, Hence, K i λ are the reproducing kernels for the pairwise orthogonal spaces S(K i λ ) with respect to the standard inner product. The convolution property (12) yields that Tr(X P λ (P, ·)) ∈ H λ (G I,d ), for any matrix Since the spaces S(K i λ ) = {0}, i = 1, . . . , |I d λ |, are orthogonally invariant and pairwise orthogonal, we obtain S(K i λ ) ∼ = H d 2λ , which yields (16).
Let K 2λ : R d×d sym × R d×d sym → R denote the reproducing kernel with respect to the differentiation inner product of the irreducible representation H d 2λ in Hom |λ| (R d×d sym ). One of the ingredients for the following proof of Theorem 5.1 is that the restriction K 2λ | G k,d ×G l,d coincides with p k,l λ up to a multiplicative constant. Indeed, we shall follow the strategy in [34]: Proof of Theorem 5.1 Consider the positive definite zonal kernel The relation (28) implies S(K ) = F λ (R d×d sym )| G d . Theorem 5.5 and (5) yield Note that v λ P λ v λ coincides up to a multiplicative factor with the restriction of the kernel K 2λ . We shall verify in the following that the kernel v λ P λ v λ reflects the expansion of p λ into zonal polynomials C λ defined in Theorem 5.1. Starting with (10), we exploit the vanishing and symmetry properties of the Jacobi polynomials, cf. Appendix B.3, combined with the symmetry relations of the intertwining functions, cf. Appendix B.2. We observe that, for any partition λ with (λ) ≤ d 2 and any (P, where b k,l,d λ ∈ R \ {0}. After inserting the expansion from (18) into both sides of (17) via P λ and P λ , we aim to compare coefficients of the zonal polynomials. Let k, l ∈ I = { (λ), . . . , d − (λ)} be fixed. One can (only) show linear independence of the functions Since λ ≤ λ implies (λ ) ≤ min(k, d − k, l, d − l), the zonal polynomials in (18) are linearly independent. By applying P λ (P, Q) = 0, (P, Q) ∈ G k,d × G l,d , for (λ ) > min(k, d − k, l, d − l), the summation on the right-hand side in (17) reduces accordingly, and comparing coefficients is justified. Hence, we In other words, we have verified that, for (λ) ≤ Tr(P), For the remaining cases, we observe that v λ P λ (P, Q)v λ = 0, and p λ (P, Q) = 0 holds due to (18).
For |λ| = 0, 1, 2, we present the kernels K 2λ and p λ as well as p k,l λ and v k λ in Appendix D. We now replace P λ with p λ and still have a suitable Fourier expansion: Proof The Fourier expansion (13) with coefficientsK k,l where v λ is given in (19). Therefore, the rank of K λ (k, l) k,l∈I d λ is the same as the one of the original Fourier coefficient, so that Theorem 5.5 implies the statement.
The following result is an important step forward: where v λ is given in (19). We can define the associated positive definite kernels K i λ : For P ∈ G k,d , Q ∈ G l,d , the identity Note that (21) also implies the lower bound on the multiplicities in Corollary 3.2.
We shall complete the proof in Appendix C, where we verify that (21) holds with equality; i.e., we decompose H |λ|+s λ (G d ) into s +1 orthogonal subspaces of increasing polynomial degree with simple multiplicities.
We are now able to determine the multiplicities of the irreducible components in Pol t (G I,d ).
where I = {k i } r i=1 is ordered such that min{k 1 , d − k 1 } ≥ · · · ≥ min{k r , d − k r }, so that equality actually holds in Theorem 3.1. Our proof of Theorem 3.1 then reveals the intriguing identity with the restriction mapping | G k 1 ,d : Pol t (G I,d ) → Pol t (G k 1 ,d ).

Cubatures and Designs on Unions of Grassmannians
So far, we have analyzed the irreducible decomposition of polynomial spaces on unions of Grassmannians. Our results enable us in the following to study cubatures on unions of Grassmannians.

Introducing Cubatures and Designs
Any orthogonally invariant finite signed measure σ I,d on G I,d is a linear combination of the Haar (probability) measures σ k,d , k ∈ I, i.e., If {(P j , ω j )} n j=1 is a cubature for Hom t (G I,d ) with k∈I m k = n j=1 ω j , then it is also a cubature for Pol t (G I,d ), cf. Theorem 4.3. The value of the parameter t is often called the strength of the cubature.  (G I,d )). For m k = 0, k ∈ I, any cubature {(P j , ω j )} n j=1 for Pol 2t (G I,d ) with respect to σ I,d with nonnegative weights satisfies n ≥ dim (Pol t (G I,d )), cf. [19,Proposition 1.7].
Analogously to Euclidean designs, cf. [7][8][9][10], cubatures on unions of Grassmannians induce cubatures on single Grassmannians, but potentially with lower strength: Proof Let f be a polynomial of degree at most s on R d×d sym ; then we know that 1 G k,d · f is a polynomial of degree at most t. Hence, the statement follows from We also observe that any cubature of strength 2t gives rise to a cubature of strength 2t + 1: Proposition 6.3 Let {(P j , ω j )} n j=1 be a cubature for Pol t (G I,d ) with respect to the signed measure σ I,d = k∈I m k σ k,d . Then Proof In view of Remark 4.4, it is sufficient to consider the polynomial Tr(X ·) t+1 | G I∪(d−I),d for X ∈ R d×d sym . Then we have Tr(X P) t+1 dσ I,d (P) Tr(X P) t+1 + (−1) t Tr(X (I − P)) t+1 dσ I,d (P).
The mapping P → Tr(X P) t+1 + (−1) t Tr(X (I − P)) t+1 is contained in Pol t (G I,d ), because the two terms with exponent t + 1 cancel out. Thus, {(P j , ω j )} n j=1 being a cubature for Pol t (G I,d ) yields Cubatures of strength t on single Grassmannians, whose weights are all the same, are called t-designs and have been studied in [2][3][4][5]. We shall extend this concept to unions of Grassmannians after a brief observation on characterstic functions. For Thus, the characteristic function of each G k,d is contained in Pol t (G I,d ), for all t ≥ |I| − 1. The latter yields that any cubature {(P j , ω j )} n j=1 for Pol t (G I,d ) with respect to σ I,d = k∈I m k σ k,d satisfies m k = { j:P j ∈G k,d } ω j , for k ∈ I. If weights are the same on each single G k,d , then those weights must be m k n k , where n k = |{ j : P j ∈ G k,d }|. Indeed, we impose this condition in our definition of designs: Definition 6.4 Let P k ⊂ G k,d be finite, and set n k := |P k |. The collection P I = k∈I P k is called a t-design with respect to an orthogonally invariant signed measure Thus, any t-design is a cubature, whose weights are the same on each Grassmannian but can differ across different Grassmannians. For |I| = 1, our definition reduces to the standard Grassmannian designs as considered in [2][3][4][5]. The existence of t-designs with t = 1 in single Grassmannians was studied in [14]. For a discussion on the existence of 1-designs in unions of Grassmannians with m k n k = m l n l , for all k, l ∈ I, we refer to [12].

Constructing Cubatures and Designs by Numerical Minimization
To construct cubatures or designs, we consider the t-fusion frame potential, cf. [6], where {P j } n j=1 ⊂ G I,d and {ω j } n j=1 . The 1-fusion frame potential was already investigated in [13,38]. Lower bounds on the t-fusion frame potential for general positive integers t were derived in [6]. Only for single Grassmannians, i.e., |I| = 1, those {(P j , ω j )} n j=1 were characterized in [6], for which the bounds are matched. In the following, we provide a characterization for the general case |I| ≥ 1. Before we state this result though, it is convenient to define where σ I,d = k∈I m k σ k,d and m = (m k ) k∈I with the matrix T I,d (t) ∈ R |I|×|I| being given by Tr(P Q) t dσ k,d (P)dσ l,d (Q).
Note that T I,d (t) is the (0)-th Fourier coefficient of the positive definite zonal kernel K t (P, Q) = Tr(P Q) t , for P, Q ∈ G I,d , and thus symmetric and positive semidefinite.
Theorem 6.5 Given {P j } n j=1 ⊂ G I,d with weights {ω j } n j=1 ⊂ R, let m k = { j:P j ∈G k,d } ω j . Then the fusion frame potential is bounded from below by Equality holds if and only if {(P j , ω j )} n j=1 is a cubature for Pol t (G I,d ) with respect to σ I,d .
This theorem is an extension of results in [6], where the lower bound is already derived but equality is not discussed. For the proof, we refer to Appendix A.2.
Theorem 6.5 enables the use of numerical minimization schemes to derive cubatures. Knowledge of the global lower bound T σ I,d (t) is important to check if numerical solutions are indeed cubatures by ruling out that the minimization got stuck in a local minimum. The matrix T I,d (t) can be computed via zonal polynomials by cf. (30) in Appendix B.1, and note that C λ (I k ) is explicitly computed in [15,30,39]. For suitable minimization algorithms on Grassmannians, we refer to [1,11,29]. In Sect. 7 we shall indeed numerically minimize the fusion frame potential with equal weights and check that the lower bound is attained.

Examples of t-Designs Derived from Numerical Minimization
Here, we shall construct some families of 1-, 2-, and 3-designs in unions of Grassmannians. By numerically minimizing the energy functional (25) using a conjugate gradient approach, cf. [29, Section 3.3.1], we compute candidates for t-designs, i.e., t-designs up to machine precision. Based on the special structures of the Gram matrices of these numerical minimizers, we looked for group orbits that describe them analytically. Indeed "beautifying" our numerical results, we were able to analytically specify our candidates, which turned out to be exact minimizers.

A Family of 1-Designs in Arbitrary Dimensions
We analytically construct d lines and 1 hyperplane in R d , so that the corresponding orthogonal projectors are a 1-design in where e i ∈ R d are the standard unit vectors and e := e 1 + · · · + e d . The associated rank-1 projectors are P i := a i a i ∈ R d , i = 1, . . . , d, and the rank-(d − 1) projector is P d+1 := I d − 1 d e e . We calculate and observe equality with the 1-fusion frame potential According to Theorem 6.5,

A Family of 1-Designs in R 4
We construct a family of 2 lines and 2 planes in R 4 forming a 1-design with respect to m 1 ∈ [−2, 2] and m 2 = 1. The rank-1 projectors are P 1 := e 1 e 1 , P 2 := e 3 e 3 ∈ R 4 , and the 2-dimensional projectors are For m 1 ∈ [−2, 2], this family provides 1-designs since equality holds in Theorem 6.5, where T m 1 σ 1,4 +σ 2,4 (1) = 1 + m 1 2 2 . For m 1 = 2, the two planes coincide and the two lines are orthogonal to each other and to the planes. The two planes also coincide for m 1 = −2, and then the two lines span the same plane. The choice m 1 = 0 yields two planes that are orthogonal to each other.
Remark 7.4 Proposition 6.2 yields that any 2-designs with d = 4, m 1 = 1, and m 2 = 3/2 must have at least 4 lines, which is matched by our example of 4 lines and 8 planes.

Remark 7.5
Similarly to the previous example, Proposition 6.2 yields that any 2-design with d = 5, m 1 = 1, and m 2 = 5/3 must have at least 5 lines, which is matched.

From 2t-Designs to 2t + 1-Designs
Theorem 6.3 yields a construction of 2t + 1-designs from 2t-designs. Hence, any 2-design in the Sects. 7.3, 7.4, and 7.5 gives rise to a 3-design. For instance, the 6 lines going through the vertices of the icosahedron are a 2-design in R 3 . By adding the 6 complementary planes, we obtain a 3-design with respect to m 1 = m 2 = 1.
Remark 7.6 Proposition 6.2 yields that the parts in each single Grassmannian G k,d of a 3-design for G I,d with |I| = 2 must be a cubature of strength 2. Any such cubature must have at least 6 elements for d = 3, so that our 3-design with 12 elements in the union of two Grassmannians is optimal.
. It turns out that the restrictions S(K )| Y to compact subsets Y ⊂ X are generated by the restricted kernel itself; i.e., if Y ⊂ X is compact, then see also [45,Theorem 10.47] for a more general setting. For more details on the theory of reproducing kernel Hilbert spaces, we refer to [45].

B.4 Even and Odd Kernels
The sets of even and odd functions on G d are denoted by respectively, and the orthogonal decomposition L 2 (G d ) = S + ⊕ S − holds. We call positive definite zonal kernels K + and K − even and odd if they satisfy The kernel K s (P, Q) := 1 + 2 Tr(P Q) s + 2 Tr((I − P)(I − Q)) s , for P, Q ∈ G d , linearly generates Pol s (G d ), cf. Remark 4.4. One observes that K 2t+1 + (P, ·) is a polynomial of degree 2t and K 2t+2 − (P, ·) is a polynomial of degree 2t + 1, cf. the proof of Proposition 6.3. Therefore, we obtain Pol 2t+2 (G d ) ∩ S − ⊂ Pol 2t+1 (G d ). (32) Hence, since H |λ| λ (G d ) has multiplicity one, its reproducing kernel p λ is an even or odd kernel, depending on |λ|; i.e., it satisfies the symmetry relations p λ (P, Q) = (−1) |λ| p λ (P, I − Q), P, Q ∈ G d , We shall verify that being even or odd transfers into symmetry conditions on the kernel's coefficients when expanded in zonal polynomials. holds.
The following proof is based on comparing coefficients in the expansion (34) after exploiting the binomial expansion of zonal polynomials.
Reproducing kernels of the subspaces of H |λ|+s λ (G I,d ) equivalent to H d 2λ with increasing polynomial degree can be derived by restrictions of the kernels K i λ in the proof of Theorem 5.7, cf. (20).