A note on EC-tractability of multivariate approximation in weighted Korobov spaces for the standard information class☆,☆☆
Section snippets
Introduction and main results
Recently, research in information-based complexity theory has focused on tractability of multivariate continuous problems. Studies on this area started in 1994 (see [18], [19]) and since then a huge number of results emerged on this topic. Nowadays tractability of multivariate problems is an active and popular research field (see [9], [10], [11] and the references therein). It has been studied in different settings including the worst case, the probabilistic case and the average case setting,
Preliminaries
For a fixed , let be the analytic Korobov kernel given by (1.5) with satisfying (1.3). By (1.4) we know that the reproducing kernel Hilbert space is a tensor product of the univariate reproducing kernel Hilbert spaces with reproducing kernels , i.e.,
Proof of part 1
For the notion of EC-QPT, observe that for and , where . In the following proof we will sometimes use these equivalent formulations to establish EC-QPT.
Proof We first prove the “only if” part. Assume that APP is EC-QPT for the class . Then by (1.8) we know that APP is also EC-QPT for the class and this implies (1.9). Furthermore, it follows from the proof of Theorem 8 in [4] that the exponent
Acknowledgements
The author would like to thank F. Pillichshammer and two anonymous referees for helpful suggestions and comments regarding an improved presentation of the results.
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Cited by (4)
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2022, International Journal of Wavelets, Multiresolution and Information Processing
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Supported by the National Natural Science Foundation of China (Grant No. 11871006) and Doctoral Foundation Project of Shandong Jianzhu University (X19090Z).
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Communicated by F. Pillichshammer.