Wavelet packets: Uniform approximation and numerical integration

2019 ◽  
Vol 18 (02) ◽  
pp. 2050004
Author(s):  
Nikhil Khanna ◽  
S. K. Kaushik ◽  
A. M. Jarrah
Keyword(s):  
Error Analysis ◽  
Numerical Integration ◽  
Uniform Approximation ◽  
Integrable Function ◽  
Wavelet Packet ◽  
Sufficient Conditions ◽  
Wavelet Packets ◽  
Definite Integral ◽  
Approach Zero ◽  
Square Integrable Function

In this paper, it is proved that under some conditions, wavelet packet basis of [Formula: see text] can be used as a tool for the uniform approximation of an [Formula: see text]-times ([Formula: see text]) continuously differentiable and square integrable function [Formula: see text]. Sufficient conditions which establish that the approximations of wavelet packet sequences of square integrable function [Formula: see text] at lower levels are uniformly reliable and they uniformly approach zero as [Formula: see text] are given. Finally, a method based on wavelet packet expansion to find the definite integral of a function in [Formula: see text] is given and its error analysis has been discussed.

scholarly journals Bounded andL2-solutions to a second order nonlinear differential equation with a square integrable forcing term

1999 ◽  
Vol 22 (3) ◽  
pp. 569-571 ◽  
Author(s):  
Allan Kroopnick
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Second Order ◽  
Forcing Term ◽  
Order Nonlinear Differential Equation ◽  
Square Integrable Function

This paper presents two theorems concerning the nonlinear differential equationx″+c(t)f(x)x′+a(t,x)=e(t), wheree(t)is a continuous square-integrable function. The first theorem gives sufficient conditions when all the solutions of this equation are bounded while the second theorem discusses when all the solutions are inL2[0,∞).

scholarly journals POSITIVE PERTURBATIONS OF SELF-ADJOINT SCHRÖDINGER OPERATORS ON RIEMANNIAN MANIFOLDS

2005 ◽  
Vol 02 (04) ◽  
pp. 543-552
Author(s):  
OGNJEN MILATOVIC
Keyword(s):  
Riemannian Manifold ◽  
Differential Expression ◽  
Riemannian Manifolds ◽  
Integrable Function ◽  
Schrödinger Operators ◽  
Sufficient Conditions ◽  
Schrodinger Operators ◽  
Square Integrable Function

We consider a Schrödinger differential expression L0 = ΔM + V0 on a Riemannian manifold (M,g) with metric g, where ΔM is the scalar Laplacian on M and V0 is a real-valued locally square integrable function on M. We consider a perturbation L0 + V, where V is a non-negative locally square-integrable function on M, and give sufficient conditions for L0 + V to be essentially self-adjoint on [Formula: see text]. This is an extension of a result of T. Kappeler.

Reconstruction of a stationary Gaussian process from its sign-changes

1977 ◽  
Vol 14 (01) ◽  
pp. 38-57
Author(s):  
Jacques de Maré
Keyword(s):  
Gaussian Process ◽  
Real Line ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Original Process ◽  
Sign Changes ◽  
Entire Real Line ◽  
Unit Variance ◽  
Square Integrable Function ◽  
Analytical Expressions

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.

Reconstruction of a stationary Gaussian process from its sign-changes

1977 ◽  
Vol 14 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Jacques de Maré
Keyword(s):  
Gaussian Process ◽  
Real Line ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Original Process ◽  
Sign Changes ◽  
Entire Real Line ◽  
Unit Variance ◽  
Square Integrable Function ◽  
Analytical Expressions

The sign-changes over the entire real line of a Gaussian process with unit variance and mean zero are observed. The Gaussian process is reconstructed by an interpolator which is optimal in the class of linear interpolators based on that process which is one when the original process is positive and minus one when it is negative. Sufficient conditions for the interpolator to be a convolution of the sign process and a square integrable function are given. Analytical expressions of the interpolation error are derived, and the behaviour of the interpolator is studied by means of simulations.

CONSTRUCTING MULTIPLE WAVELET PACKETS WITH FIFS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 165-169
Author(s):  
GANG CHEN ◽  
ZHIGANG FENG
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Scaling Functions ◽  
Wavelet Packet Decomposition ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

By using fractal interpolation functions (FIF), a family of multiple wavelet packets is constructed in this paper. The first part of the paper deals with the equidistant fractal interpolation on interval [0, 1]; next, the proof that scaling functions ϕ1, ϕ2,…,ϕr constructed with FIF can generate a multiresolution analysis of L2(R) is shown; finally, the direct wavelet and wavelet packet decomposition in L2(R) are given.

scholarly journals Minimization of Harmonics Noise Using Wavelet Transformation Technology

2011 ◽  
pp. 216-223
Author(s):  
Mr. Debasis Dash ◽  
Mr. Shatyaprakasha Satapathy ◽  
Dr. Chittaranjan Panda
Keyword(s):  
Wavelet Analysis ◽  
Wavelet Packet ◽  
Statistical Significance ◽  
Audio Signal ◽  
Design Procedure ◽  
Wavelet Packets ◽  
Threshold Function ◽  
Analysis Tool ◽  
Real Time Processing ◽  
Wavelet Spectra

The field programmable gate array technology can design high performance system at low cost for wavelet analysis. Wavelet transform has gained the reputation of being a very effective signal analysis tool for much practical application. Implementation of transform needs the meeting of real-time processing for most application. The objectives of this paper are to compare the Haar and Daubeches technology and to calculate the bit error rate (BER) between the input audio signal and reconstructed output signal. It is seen that the BER using Daubechies wavelet technology is less than Haar wavelet. The design procedure is explained using the stat of art electronic design. Automation tools for system design on FPGA, simulation, synthesis and implementation on the FPGA technology has been carried out. The power hovmoller, cross wavelet spectra and coherence are described. A Practical step-up-step guide to wavelet analysis is given with examples taken from time series. The guide includes a comparison to the windowed Fourier transform. New statistical significance test for wavelet power spectra are developed by deriving theoretical wavelet spectra for white and red noise. Empirical formula is given for the effect of smoothing on significance levels and filtering. The notion of orthogonal no separable trivet wavelet packets, which is the generation of orthogonal university wavelet packets is introduced. A de-noising method based on wavelet packet shrinkage is developed. The principle of wavelet packet shrinkage for de-noising and the section of thresholds and threshold function are analyzed.

scholarly journals What is the Wigner Function Closest to a Given Square Integrable Function?

2018 ◽  
Vol 50 (5) ◽  
pp. 5161-5197 ◽  
Author(s):  
J. S. Ben-Benjamin ◽  
L. Cohen ◽  
N. C. Dias ◽  
P. Loughlin ◽  
J. N. Prata
Keyword(s):  
Wigner Function ◽  
Integrable Function ◽  
Square Integrable Function

scholarly journals On Generalized Carleson Operators of Periodic Wavelet Packet Expansions

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Shyam Lal ◽  
Manoj Kumar
Keyword(s):  
Wavelet Packet ◽  
Wavelet Packets ◽  
Arithmetic Means ◽  
Periodic Wavelet

Three new theorems based on the generalized Carleson operators for the periodic Walsh-type wavelet packets have been established. An application of these theorems as convergence a.e. for the periodic Walsh-type wavelet packet expansion of block function with the help of summation by arithmetic means has been studied.

The Energy Distribution of Tire-Road Noise and Noisy Speech Based on Wavelet Packet

2011 ◽  
Vol 464 ◽  
pp. 721-724 ◽  
Author(s):  
Zhi Yong He ◽  
Li Heng Luo
Keyword(s):  
Energy Distribution ◽  
Speech Enhancement ◽  
Mobile Communications ◽  
Wavelet Packet ◽  
Noise Signal ◽  
Wavelet Packets ◽  
Road Noise ◽  
Noisy Speech ◽  
Time Frequency ◽  
The Difference

Speech enhancement is very important for mobile communications or some other applications in car. The energy distribution of signal is the basis of algorithms which denoise noisy speech in time-frequency domain. In this work, the noise regarded is the tire-road noise when driving in expressway. Wavelet packets transform is used in the analysis. After decomposing noise signal and noisy speech signal by wavelet packet transform, the analysis for the difference of the energy distribution between noisy speech and noise is finished.

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