Abstract
We utilize moduli of smoothness and \(K\)-functionals as new tools in the arena of estimating the decay rates of eigenvalue sequences associated with some commonly used positive integral operators on spheres. This approach is novel and effective. We develop two readily verifiable and implementable conditions for the kernels of the integral operators under which favorable decay rates of eigenvalue sequences are derived. The first one (based on spherical mean operators) is an enhancement of the classical Hölder condition. The second one, works seamlessly with the Laplace-Beltrami operators and can be applied directly to Bessel potential kernels.
Keywords
- Sphere
- Decay rates
- Positive integral operators
- Fourier coefficients
- Moduli of smoothness
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Notes
- 1.
Kühn and his co-author studied integrated Hölder conditions in Cobos and Kühn [4].
- 2.
Here we call \(K\) a Bessel potential kernel if \(K\) can reproduce a Bessel potential Sobolev space on spheres. We refer readers to Mhaskar et al. [15] for technical details in this regard.
- 3.
The Laplace-Betrami operator is the restriction to the sphere \(S^m\) of the classical Laplace operator
$$ \Delta = \frac{\partial ^2}{\partial x_1^2} + \cdots + \frac{\partial ^2}{\partial x_{m+1}^2} $$in the Euclidean space \(\mathbb {R}^{m+1}\).
- 4.
The shifting operator here can be considered as the restriction to \(S^m\) of the spherical mean operator in \(\mathbb {R}^{m+1}\).
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Acknowledgments
The first author was partially supported by FAPESP, grant \(\#\) 2012/25097-4. Part of this research was done while the first author visited the Department of Mathematics at Missouri State University. She is thankful to many people in the host institution for their hospitality. We are grateful to two anonymous referees for their suggestions that have enhanced the exposition of the paper.
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Jordão, T., Menegatto, V.A., Sun, X. (2014). Eigenvalue Sequences of Positive Integral Operators and Moduli of Smoothness. In: Fasshauer, G., Schumaker, L. (eds) Approximation Theory XIV: San Antonio 2013. Springer Proceedings in Mathematics & Statistics, vol 83. Springer, Cham. https://doi.org/10.1007/978-3-319-06404-8_13
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