Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere

Jordao, T.; Menegatto, V. A.

doi:10.1090/proc12716

Cited by 9 publications

(5 citation statements)

References 21 publications

(28 reference statements)

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“…Example 2.3. (Averages on caps) This example is discussed in [2,6], while the point of view we will give here is aligned with [9]. The average operator on the cap C x t = {w ∈ S m : x • y ≥ cos t} of S m , defined by t, is the operator A t given by…”

Section: Example 22 (Shifting Operator)mentioning

confidence: 99%

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

Jordão¹,

Menegatto²

2019

Results Math

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Approximation processes in the reproducing kernel Hilbert space associated to a continuous kernel on the unit sphere S m in the Euclidean space R m+1 are known to depend upon the Mercer's expansion of the compact and self-adjoint L 2 (S m )-operator associated to the kernel. The estimation of the Kolmogorov n-th width of the unit ball of the reproducing kernel Hilbert space in L 2 (S m ) and the identification of the so-called optimal subspace usually suffice. These Kolmogorov widths can be computed through the eigenvalues of the integral operator associated to the kernel. This paper provides sharp upper bounds for the Kolmogorov widths in the case in which the kernel satisfies an abstract Hölder condition. In particular, we follow the opposite direction usually considered in the literature, that is, we estimate the widths from decay rates for the sequence of eigenvalues of the integral operator.

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Section: Example 22 (Shifting Operator)mentioning

confidence: 99%

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

Jordão¹,

Menegatto²

2019

Results Math

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“…The following theorem relates the decay of the Fourier coefficients of a function to the rate of approximation of operator defined in formula (1.4). In [11] a proof of similar result is presented for a multiplier operator on the spherical setting and it is slightly different from below.…”

Section: Decay Of Fourier Coefficientsmentioning

confidence: 99%

On approximation tools and its applications on compact homogeneous spaces

Carrijo¹,

Jordão²

2017

Preprint

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We prove a characterization for the Peetre type K-functional on M, a compact twopoint homogeneous space, in terms the rate of approximation of a family of multipliers operator defined to this purpose. This extends the well known results on the spherical setting. The characterization is employed to show that an abstract Hölder condition or finite order of differentiability condition imposed on kernels generating certain operators implies a sharp decay rates for their eigenvalues sequences. The latest is employed to obtain estimates for the Kolmogorov n-width of unit balls in Reproducing Kernel Hilbert Space (RKHS).

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“…Recently, we have developed a new technique to deduce sharp decay rates for the sequence of eigenvalues of positive integral operators based on growth and integrability of Fourier coefficients ( [8,9]). This technique allows one to work in an even more general setting, replacing all the arguments involving the usual spherical convolutions with that of spherical convolutions with measures.…”

Section: Final Remarksmentioning

confidence: 99%

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

Jordao¹,

Menegatto²

2015

Mathematical Inequalities &Amp; Applications

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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.

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Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere

Cited by 9 publications

References 21 publications

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

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

On approximation tools and its applications on compact homogeneous spaces

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

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