Jackson kernels: a tool for analysing the decay of eigenvalue sequences of integral operators on the sphere

Abstract

Abstract. Decay rates for the sequence of eigenvalues of positive and compact integral operators have been largely investigated for a long time in the literature. In this paper, the focus will be on positive integral operators acting on square integrable functions on the unit sphere and generated by a kernel satisfying a Hölder type assumption defined by average operators. In the approach to be presented here, the decay rate will be reached from convenient estimations on the eigenvalues of the operator themselves, with the help of specific properties of a generic approximation operator defined through the so-called generalized Jackson kernels. The decay rate has the same structure of those known to hold in the cases in which the Hölder condition is the classical one. Therefore, within the spherical setting, the abstract approach to be introduced here extends some classical results on the topic.Mathematics subject classification (2010): 41A36, 45P05, 47A75, 47B34.

Kolmogorov Widths on the Sphere via Eigenvalue Estimates for Hölderian Integral Operators

“…where supp(K t (x, ·)) is the support of the function y ∈ S m → K t (x, y) = K t i (x · y). Observe that supp(K t (x, ·)) does not depend upon x due to the invariance of σ m with respect to orthogonal transformations on R m+1 ( [10]).…”

“…If G ′ n = J l,n , Proposition 2 in [10] reveals that A n,1 has rank at most d m+1 ln+1 . Regarding Example 2.4, the family of measures {µ Wt : t ∈ (0, π)} also fits into the setting of Theorem 3.3 with c k,m as in the previous case, v m (t) = D m (t) for all t, α(k) = k − 1, β = (m + 1)/2, and γ = m. If G ′ n = J l,n , Proposition 3 in [10] reveals that the operator A n,m has rank at most d m+1 ln+1 .…”

“…with C > 0.This is the point where we need a special choice of the integer l in order to proceed. Picking l so that 2l ≥ ρ + c(m) + 2β − γ + 1 (c(m) is the constant in (4.11)), we can apply Lemma 1 in [10] in order to see that…”

“…Prior to that, we recall a technical lemma involving the square root K 1/2 of the operator K. We use • HS to denote the usual Hilbert-Schmidt norm of a Hilbert-Schmidt operator. Its proof can be easily adapted from Lemmas 4 and 5 in [10].…”