2015
DOI: 10.7153/mia-18-115
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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 themsel… Show more

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Cited by 1 publication
(4 citation statements)
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“…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]).…”
Section: Approximation Operatorsmentioning
confidence: 99%
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“…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]).…”
Section: Approximation Operatorsmentioning
confidence: 99%
“…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 .…”
Section: It Follows Thatmentioning
confidence: 99%
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