Best kernel approximation in Bergman spaces

Qu, Wei; Qian, Tao; Li, Haichou; Zhu, Kehe

doi:10.1016/j.amc.2021.126749

Cited by 3 publications

(2 citation statements)

References 21 publications

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“…For the self-containing purpose this section will introduce adaptive Fourier decomposition (AFD) type sparse representations with emphasis on stochastic AFDs ( [24,13]). We are based on a dictionary D of a complex Hilbert space H. By definition, a dictionary of H consists of a class of unimodular elements whose linear span is dense in H. The formulation we adopt is that H is the L 2 -space of complex-valued functions on a manifold ∂D, being the boundary of D, where D itself is an open and connected domain, called a region, in an Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

“…The existence result for the classical complex Hardy space case has recently been re-proved by using a new approach based on the maximal module principle of complex analytic functions ( [31]). This progress allows to generalize the existence of an n-best solution to weighted Bergman spaces ( [24]), and further to a wide class of RKHSs ( [17]): Under a set of commonly used conditions existence of an n-best approximation is proved for a large class of RKHSs consisting of certain analytic functions in the unit disc. In the upper-half of the complex plane there is a parallel theory.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

Tao¹,

Qu²

2022

Preprint

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This article introduces adaptive Fourier decomposition (AFD) type methods, emphasizing on those that can be applied to stochastic processes and random fields, mainly including stochastic adaptive Fourier decomposition and stochastic pre-orthogonal adaptive Fourier decomposition. We establish their algorithms based on the covariant function and prove that they enjoy the same convergence rate as the Karhunen-Loève (KL) decomposition. The AFD type methods are compared with the KL decomposition. In contrast with the latter, the AFD type methods do not need to compute eigenvalues and eigenfunctions of the kernelintegral operator induced by the covariance function, and thus considerably reduce the computation complexity and computer consumes. Various kinds of dictionaries offer AFD flexibility to solve problems of a great variety, including different types of deterministic and stochastic equations. The conducted experiments show, besides the numerical convenience and fast convergence, that the AFD type decompositions outperform the KL type in describing local details, in spite of the proven global optimality of the latter.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

Tao¹,

Qu²

2022

Preprint

Self Cite

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n-Best kernel approximation in reproducing kernel Hilbert spaces

Qian

2023

Applied and Computational Harmonic Analysis

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Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

Song

Sun

2022

Mathematics

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The central problem of this study is to represent any holomorphic and square integrable function on the Kepler manifold in the series form based on Fourier analysis. Because these function spaces are reproducing kernel Hilbert spaces (RKHS), three different domains on the Kepler manifold are considered and the weak pre-orthogonal adaptive Fourier decomposition (POAFD) is proposed on the domains. First, the weak maximal selection principle is shown to select the coefficient of the series. Furthermore, we prove the convergence theorem to show the accuracy of our method. This study is the extension of work by Wu et al. on POAFD in Bergman space.

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Best kernel approximation in Bergman spaces

Cited by 3 publications

References 21 publications

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

AFD Types Sparse Representations vs. the Karhunen-Loeve Expansion for Decomposing Stochastic Processes

n-Best kernel approximation in reproducing kernel Hilbert spaces

Representing Functions in H2 on the Kepler Manifold via WPOAFD Based on the Rational Approximation of Holomorphic Functions

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