Introduction to Fourier Analysis and Wavelets

Pinsky, Mark A.

doi:10.1090/gsm/102

Cited by 158 publications

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“…Remarkably, the coefficients presented in (1) reduce, in this limit, to elegant trigonometric series [9] for the periodicized monomials -exactly! - Indeed, for finite j, the coefficients are nothing but the Taylor polynomial truncations of these same trigonometric series.…”

Section: Discussionmentioning

confidence: 99%

A Compact Formula for Rotations as Spin Matrix Polynomials

Curtright

Fairlie

Zachos

2014

SIGMA

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Section: Discussionmentioning

confidence: 99%

A Compact Formula for Rotations as Spin Matrix Polynomials

Curtright

Fairlie

Zachos

2014

SIGMA

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“…An admissible wavelet ψ satisfies The wavelet transform of x is then the inner product Here t is the point of interest in the signal and s is the positive scale at which the wavelet is dilated. The admissibility condition enables the reconstruction of the original function from its wavelet transform in suitable function spaces, see for example [4]. To make this definition useful for the theory and also for the actual vibration signals, x must be from the class of generalised functions called tempered distributions.…”

Section: Continuous Wavelet Transformmentioning

confidence: 99%

Local Regularity Analysis with Wavelet Transform in Gear Tooth Failure Detection

Nissilä

2017

Management Systems in Production Engineering

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Abstract:Diagnosing gear tooth and bearing failures in industrial power transition situations has been studied a lot but challenges still remain. This study aims to look at the problem from a more theoretical perspective. Our goal is to find out if the local regularity i.e. smoothness of the measured signal can be estimated from the vibrations of epicyclic gearboxes and if the regularity can be linked to the meshing events of the gear teeth. Previously it has been shown that the decreasing local regularity of the measured acceleration signals can reveal the inner race faults in slowly rotating bearings. The local regularity is estimated from the modulus maxima ridges of the signal's wavelet transform. In this study, the measurements come from the epicyclic gearboxes of the Kelukoski water power station (WPS). The very stable rotational speed of the WPS makes it possible to deduce that the gear mesh frequencies of the WPS and a frequency related to the rotation of the turbine blades are the most significant components in the spectra of the estimated local regularity signals

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“…You can use continuous parameter or discrete parameter space {L n , l n }. As it is a work that can be carried to the transformed frame, the measure and integral within will be defined according to Lebesgue [2] [15] and the analysis can be performed in discrete space.…”

Section: Definition 4-the Quality Indexmentioning

confidence: 99%

“…The natural harmony of Fourier's analysis in the Hilbert space is demonstrated by Riesz-Fischer's [1] and Plancherel's [2] theorems. In this harmony, three concepts are summarized: reflexibility, regularity and series obtention.…”

Section: Introductionmentioning

confidence: 99%

Wavelets and Entropy for Power Quality Assessment

Plata¹,

Ustariz-Farfan²,

Estrada-Estrada³

2017

AJCM

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In this paper, wavelet transform and entropy are evaluated using the mathematical analysis concepts of reflexibility, regularity and series obtention, these concepts remark the reason to make a selective reference framework for power quality applications. With this idea the paper used the same treatment for the two algorithms (Multiresolution and Multiscale Entropy). The wavelet is denoted to have the most power full consistence to the light off the reflexibility, regularity and series obtention. The paper proposes a power quality technique namely M pq AT.

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Introduction to Fourier Analysis and Wavelets

Cited by 158 publications

References 0 publications

A Compact Formula for Rotations as Spin Matrix Polynomials

A Compact Formula for Rotations as Spin Matrix Polynomials

Local Regularity Analysis with Wavelet Transform in Gear Tooth Failure Detection

Wavelets and Entropy for Power Quality Assessment

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