scholarly journals Construction of a family of non-stationary biorthogonal wavelets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Baoxing Zhang ◽  
Hongchan Zheng ◽  
Jie Zhou ◽  
Lulu Pan
Keyword(s):  
Multiresolution Analysis ◽  
Spline Functions ◽  
Refinable Functions ◽  
Biorthogonal Wavelet ◽  
Biorthogonal Wavelets ◽  
Vanishing Moments ◽  
B Spline ◽  
The Family ◽  
Cardinal Functions ◽  
The Stability

Abstract The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.

scholarly journals Multiresolution analysis and wavelets on Vilenkin groups

2008 ◽  
Vol 21 (3) ◽  
pp. 309-325 ◽  
Author(s):  
Yury Farkov
Keyword(s):  
Multiresolution Analysis ◽  
Linear Independence ◽  
Sufficient Conditions ◽  
Partition Of Unity ◽  
Necessary And Sufficient Conditions ◽  
Refinable Functions ◽  
Vilenkin Groups ◽  
Orthogonal Wavelets ◽  
The Stability ◽  
Necessary And Sufficient

This paper gives a review of multiresolution analysis and compactly sup- ported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of 'integer shifts' of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .

scholarly journals Fractional nonlinear Volterra–Fredholm integral equations involving Atangana–Baleanu fractional derivative: framelet applications

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mutaz Mohammad ◽  
Alexander Trounev
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Numerical Approximation ◽  
Fredholm Integral Equations ◽  
Spline Functions ◽  
Vanishing Moments ◽  
B Spline ◽  
B Splines ◽  
Cas Wavelets

Abstract In this work, we propose a framelet method based on B-spline functions for solving nonlinear Volterra–Fredholm integro-differential equations and by involving Atangana–Baleanu fractional derivative, which can provide a reliable numerical approximation. The framelet systems are generated using the set of B-splines with high vanishing moments. We provide some numerical and graphical evidences to show the efficiency of the proposed method. The obtained numerical results of the proposed method compared with those obtained from CAS wavelets show a great agreement with the exact solution. We confirm that the method achieves accurate, efficient, and robust measurement.

scholarly journals New trigonometric B-spline approximation for numerical investigation of the regularized long-wave equation

Open Physics ◽  
2021 ◽  
Vol 19 (1) ◽  
pp. 758-769
Author(s):  
Ahmed Hussein Msmali ◽  
Mohammad Tamsir ◽  
Neeraj Dhiman ◽  
Mohammed A. Aiyashi
Keyword(s):  
Applied Sciences ◽  
Spline Approximation ◽  
Spline Functions ◽  
Spline Collocation ◽  
B Spline ◽  
Long Wave ◽  
Collocation Technique ◽  
Regularized Long Wave Equation ◽  
Spatial Derivatives ◽  
The Stability

Abstract The objective of this work is to propose a collocation technique based on new cubic trigonometric B-spline (NCTB-spline) functions to approximate the regularized long-wave (RLW) equation. This equation is used for modelling numerous problems occurring in applied sciences. The NCTB-spline collocation method is used to integrate the spatial derivatives. We use the Rubin–Graves linearization technique to linearize the non-linear term. The accuracy and efficiency of the technique are examined by employing it on three important numerical examples which have three invariants of motion viz. mass, momentum, and energy. It is observed that the error norms of the present method are less than the error norms of the methods available in the literature. The numerical values of these invariants have also been approximated, which remain conserved during the program run which shows that the propagation of the solitary wave is represented perfectly. The propagation of one and two solitary waves and undulations of waves are depicted graphically. The stability analysis shows that the method is unconditionally stable.

scholarly journals Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Owais Ahmad ◽  
Neyaz A. Sheikh ◽  
Mobin Ahmad
Keyword(s):  
Multiresolution Analysis ◽  
Linear Span ◽  
Complete Characterization ◽  
Riesz Bases ◽  
Half Line ◽  
Scaling Functions ◽  
Biorthogonal Wavelet ◽  
Biorthogonal Wavelets ◽  
Single Function ◽  
Positive Half

AbstractIn this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

scholarly journals Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

2011 ◽  
Vol 2011 ◽  
pp. 1-17 ◽  
Author(s):  
Yeon Ju Lee ◽  
Jungho Yoon
Keyword(s):  
Riesz Bases ◽  
Refinable Functions ◽  
Biorthogonal Wavelet ◽  
Wavelet Bases ◽  
Refinable Function ◽  
B Splines ◽  
Compactly Supported ◽  
Mathematical Properties ◽  
The Stability

This paper is concerned with analyzing the mathematical properties, such as the regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines. We first discuss the biorthogonality condition of the nonstationary refinable functions, and then we show that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of a refinable function. For this reason, we prove that the suggested nonstationary wavelets form Riesz bases for the space that they generate.

Use of an Ambulatory Infusion Pump in a 12-Year-Old with Salmonella Osteomyelitis

DICP ◽  
1989 ◽  
Vol 23 (5) ◽  
pp. 379-381
Author(s):  
Allen J. Vaida ◽  
Chantel J. Mattiucci ◽  
Steven A. Shapiro ◽  
Linda A. Gusenko ◽  
Anna M. King
Keyword(s):  
Cost Saving ◽  
Antibiotic Therapy ◽  
Infusion Pump ◽  
Medication Administration ◽  
Sterile Water ◽  
Home Therapy ◽  
Parenteral Antibiotic ◽  
The Family ◽  
The Stability ◽  
Salmonella Osteomyelitis

A 12-year-old girl with sickle cell hemoglobinopathy presented with a Salmonella osteomyelitis of her right humerus requiring six weeks of parenteral antibiotic therapy. Home therapy was evaluated. Due to the frequency of the medication administration (every six hours) and the apprehension of the family members, a Pharmacia-Deltec CADD-VT Infusion Pump was chosen for drug administration. Based on the stability of ampicillin, 1.3 g q6h was administered to provide a minimum of 1 g for the last dose of a 24-hour cycle. Ampicillin 6 g contained in 100 mL of sterile water for injection provided a 60 mg/mL solution with an osmolarity of 347 mOsmol. The pump was programmed to deliver 22 mL of solution over one hour, every six hours. A keep-vein-open rate of 0.2 mL/h maintained line patency. A 100 mL cassette of solution prepared daily was replaced on the pump by a home therapy nurse each morning. At the end of six weeks of therapy, the osteomyelitis was eradicated. We found the use of an ambulatory infusion pump an effective, convenient, and cost-saving method of treatment for our patient.

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