Frames for operators in Banach spaces via semi-inner products

Dastourian, Bahram; Janfada, Mohammad

doi:10.1142/s0219691316500119

Cited by 6 publications

(4 citation statements)

References 33 publications

(38 reference statements)

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“…To perform signal analysis using WPD, it is essential to define the decomposition levels and select the appropriate filtering waveforms (such as db1, db2, db4, haar, etc.) for extracting detail and approximation coefficients at each level (Dastourian, Dastourian, Dastourian, & Mahnaie, 2014). Once the signal analysis is completed, the data organization phase begins in preparation for the prediction model.…”

Section: Data Pre-processing Using the Wavelet Packet Decompositionmentioning

confidence: 99%

ANFIS-based Framework for the Prediction of Bearing’s Remaining Useful Life

Lourari,

Benkedjouh,

El Yousfi

et al. 2024

IJPHM

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Bearings are critical components extensively used in rotary machines, often being the leading cause of unexpected machine shutdowns. To mitigate system failures, it is crucial to implement effective maintenance strategies. This paper introduces a novel methodology for bearing prognostics, employing Wavelet Packet Decomposition (WPD) for data preprocessing, Sequential Backward Selection (SBS) for feature selection, and Adaptive Neuro-Fuzzy Inference System (ANFIS) networks for prognostic modeling. The proposed approach consists of two key steps. Firstly, the data undergoes preprocessing through Wavelet Packet Decomposition, enhancing the quality and extracting relevant features. Subsequently, the Remaining Useful Life (RUL) of the bearing is predicted using a degradation model. The accuracy of the proposed method is evaluated using a bearing life dataset obtained from a run-to-failure test (IMS dataset). The results demonstrate the remarkable capability of the ANFIS model to learn and accurately estimate the system’s RUL. By leveraging the combined power of WPD, SBS, and ANFIS, this methodology showcases its potential as an effective prognostic tool for bearing health assessment and proactive maintenance planning.

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Section: Data Pre-processing Using the Wavelet Packet Decompositionmentioning

confidence: 99%

ANFIS-based Framework for the Prediction of Bearing’s Remaining Useful Life

Lourari,

Benkedjouh,

El Yousfi

et al. 2024

IJPHM

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“…Since then, various RKBSs [13,16,38,39,44,47,48,50] have been constructed for different applied and theoretical purposes. They are used in a wide variety of fields such as machine learning [44,47,48,50,51], sampling reconstruction [14,15,18,27,28], sparse approximation [38,39,44,47], and functional analysis [9,36,49]. The definitions of existing RKBSs and the associated reproducing kernels in the literature are dependent on the construction.…”

Section: Introductionmentioning

confidence: 99%

On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

Lin¹,

Zhang²,

Zhang³

2019

Preprint

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Recently, there has been emerging interest in constructing reproducing kernel Banach spaces (RKBS) for applied and theoretical purposes such as machine learning, sampling reconstruction, sparse approximation and functional analysis. Existing constructions include the reflexive RKBS via a bilinear form, the semi-inner-product RKBS, the RKBS with ℓ 1 norm, the p-norm RKBS via generalized Mercer kernels, etc. The definitions of RKBS and the associated reproducing kernel in those references are dependent on the construction. Moreover, relations among those constructions are unclear. We explore a generic definition of RKBS and the reproducing kernel for RKBS that is independent of construction. Furthermore, we propose a framework of constructing RKBSs that unifies existing constructions mentioned above via a continuous bilinear form and a pair of feature maps. A new class of Orlicz RKBSs is proposed. Finally, we develop representer theorems for machine learning in RKBSs constructed in our framework, which also unifies representer theorems in existing RKBSs.

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“…Poumai and Jahan [30] introduced atomic systems for operators in Banach spaces. Frames for operators in Banach spaces were further studied in [8,18]. Outline of the paper.…”

Section: Introductionmentioning

confidence: 99%

“…Gavruta [15], introduced and studied atomic system for an operator K and the notion of K-frame in a Hilbert space, see also [16]. Frames for operators in Banach spaces were further studied in [8,17,25]. Xiao et al [32] discussed relationship between K-frames and ordinary frames in Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

Approximative K-atomic decompositions and frames in Banach spaces

Jahan

2018

AJMS

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K-frames and atomic systems for an operator K in Hilbert spaces were introduced by Gavruta [17] and further studied by Xio, Zhu and Gavruta [35]. In this paper, we have introduced the notion of an approximative atomic system for an operator K in Banach spaces and obtained interesting results. A complete characterization of family of approximative local atoms of subspace of Banach space has been obtained. Also, a necessary and sufficient condition for the existence of an approximative atomic system for an operator K is given. Finally, explicit methods are given for the construction of an approximative atomic systems for an operator K from a given Bessel sequence and approximative X d -Bessel sequence.∞ k=1 f, f k f k , f ∈ H. The frame operator S is bounded, linear and invertible on H. Thus, a frame for H allows each vector in H to be written as a linear combination of the elements in the frame, but the linear independence between the elements is not required; i.e for each vector f ∈ H we have,

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Frames for operators in Banach spaces via semi-inner products

Cited by 6 publications

References 33 publications

ANFIS-based Framework for the Prediction of Bearing’s Remaining Useful Life

ANFIS-based Framework for the Prediction of Bearing’s Remaining Useful Life

On Reproducing Kernel Banach Spaces: Generic Definitions and Unified Framework of Constructions

Approximative K-atomic decompositions and frames in Banach spaces

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