Metric properties in the mean of polynomials on compact isotropy irreducible homogeneous spaces

Abstract

Let M = G/H be a compact connected isotropy irreducible Riemannian homogeneous manifold, where G is a compact Lie group (may be, disconnected) acting on M by isometries. This class includes all compact irreducible Riemannian symmetric spaces and, for example, the tori R n /Z n with the natural action on itself extended by the finite group generated by all permutations of the coordinates and inversions in circle factors. We say that u is a polynomial on M if it belongs to some G-invariant finite dimensional subspace E of L 2 (M ). We compute or estimate from above the averages over the unit sphere S in E for some metric quantities such as Hausdorff measures of level set and norms in L p (M ), 1 ≤ p ≤ ∞, where M is equipped with the invariant probability measure. For example, the averages over S of u L p (M ) , p ≥ 2, are less than p+1 e independently of M and E . * Part of the work was done during my stay in the Institut Mittag-Leffler (Djursholm, Sweden), 2011 fall. I thank the Institut for support and hospitality.

The Kostlan–Shub–Smale random polynomials in the case of growing number of variables

“…The proof of (30) is standard. Since σ a is a critical point of ψ a and ψ a (σ a ) = 1 by (26), we have ln ψ a (σ a + τ ) = ατ 2 For ξ a we have ln ξ a (σ a + τ ) = βτ…”

“…The Kostlan-Shub-Smale model for random polynomials is the Gaussian distribution in P n whose density is proportional to e − |x| 2 for some special Euclidean norm | | (see Section 2 for the definition). Expectations of some metric quantities of random functions from finite dimensional Euclidean shift-invariant function spaces on compact isotropy irreducible homogeneous Riemannian manifolds were considered in [2]. Let M and E be such a homogeneous space and a function space, respectively.…”

Decomposition of the Kostlan–Shub–Smale model for random polynomials

“…Since M is isotropy irreducible, the invariant Riemannian metric on it is unique up to a scaling factor. Hence it is the quotient of some bi-invariant metric on G. This implies that any G-invariant finite dimensional function space on M is ∆ Minvariant (the introduction in [10] contains more details; for M = S m this is true because the summands in 1.1 are irreducible and pairwise non-equivalent). Hence the summands of the G-invariant orthogonal decomposition (1.6) are eigenspaces of ∆ M .…”

“…and, moreover, s2 = , where λ is the eigenvalue of −∆ M on E (in [10], it is assumed that E ⊥ 1 but for the space of constant functions the equality s = λ m is evidently true since λ = s = 0). Since the invariant inner products on irreducible G-modules are pairwise proportional, the arguments above prove the first three equalities.…”

“…Since M is isotropy irreducible, any G-irreducible component of E is an eigenspace of ∆ M . In general, s depends on the irreducible components of E and the inner product in E. For the norm of L 2 (M ) in E the answer is given in [10,Lemma 1]: if…”