We introduce the notion of reproducing kernel banach spaces (RKBS) and study special semi-inner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its reproducing kern...
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We introduce the notion of reproducing kernel banach spaces (RKBS) and study special semi-inner-product RKBS by making use of semi-inner-products and the duality mapping. Properties of an RKBS and its reproducingkernel are investigated. As applications, we develop in the framework of RKBS standard learning schemes including minimal norm interpolation, regularization network, support vector machines, and kernel principal component analysis. In particular, existence, uniqueness and representer theorems are established.
Given a countable subset of a set , we investigate the construction of all the reproducingkernel Hilbert (resp. banach) spaces on that have as a sampling (resp. p-sampling) set. Unlike the general p-frames, we prove ...
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Given a countable subset of a set , we investigate the construction of all the reproducingkernel Hilbert (resp. banach) spaces on that have as a sampling (resp. p-sampling) set. Unlike the general p-frames, we prove that every p-sampling set for a reproducingkernelbanach space yields a reconstruction formula. Some applications are given to demonstrate the general construction.
The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems asso...
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The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. Besides, it has been the cornerstone for a significant mathematical literature on the topic of sampling theorems associated with differential and difference problems. In this work we provide, in an unified way, new and old generalizations of this result corresponding to various different settings;all these generalizations are illustrated with examples. All the different situations along the paper share a basic approach: the functions to be sampled are obtaining by duality in a separable Hilbert space through an -valued kernel K defined on an appropriate domain.
We view regularized learning of a function in a banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel banach spaces, we prove the representer theorem for the minim...
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We view regularized learning of a function in a banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.
We view regularized learning of a function in a banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel banach spaces, we prove the representer theorem for the minim...
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We view regularized learning of a function in a banach space from its finite samples as an optimization problem. Within the framework of reproducing kernel banach spaces, we prove the representer theorem for the minimizer of regularized learning schemes with a general loss function and a nondecreasing regularizer. When the loss function and the regularizer are differentiable, a characterization equation for the minimizer is also established.
Frames in a banach space B were defined as a sequence in its dual space B* in some recent references. We propose to define them as a collection of elements in B by making use of semi-inner products. Classical theory o...
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Frames in a banach space B were defined as a sequence in its dual space B* in some recent references. We propose to define them as a collection of elements in B by making use of semi-inner products. Classical theory on frames and Riesz bases is generalized under this new perspective. We then aim at establishing the Shannon sampling theorem in banachspaces. The existence of such expansions in translation invariant reproducingkernel Hilbert and banachspaces is discussed. (C) 2010 Elsevier Inc. All rights reserved.
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