On the Decay Rate of Singular Values of Integral Operators

Levine, Matthew

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On the Decay Rate of Singular Values of Integral Operators

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Integral operators are ubiquitous in all areas of pure and applied mathematics, as well as in modeling in population biology, wave propagation theory, mechanical engineering, and image compression and deblurring. The simplest argument explaining the omnipresence of integral operators is that an ordinary differential equation is equivalent to an integral equation, and this provides the basis for the classic proof of the Picard-Lindelof theorem for existence and uniqueness of solutions. Regarding partial differential equations, integral operators appear as fundamental solutions. Integral operators are also instrumental in applications such as image compression and deblurring and other more general inverse problems. The decay rate of the singular values of integral operators is crucial to building computational inversions. Indeed, this decay rate is intimately related to the dilation parameter in Tykhonov regularization and truncations of singular vector expansions. This decay rate is intimately related to the regularity properties of the integration kernel. In dimension one, this relation is well understood and can be analyzed using relatively elementary integral operator theory tools. In this thesis, we revisit the convergence rate proofs given in Chung-Wei Ha's "Eigenvalues of Differentiable Positive Definite Kernels" (their work is also based on T. T. Kadota's "Term-by-Term Differentiability of Mercer's Theorem"). We found that some arguments in are too succinct and hard to grasp for a graduate student or just someone with limited familiarity in this field. We provide additional explanations and a few more lemmas to make these arguments more accessible. In addition, we explore how these arguments could be extended to higher dimensions. We explain why this is a non-trivial endeavor; it will be the subject of future work.

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