scholarly journals Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
A. San Antolín ◽  
R. A. Zalik
Keyword(s):  
Closed Form ◽  
Spectral Factorization ◽  
Extension Principle ◽  
Refinable Functions ◽  
Wavelet Frames ◽  
Compactly Supported ◽  
Slight Generalization ◽  
Dilation Matrix ◽  
Transcendental Functions ◽  
Low Degrees

For any dilation matrix with integer entries and , we construct a family of smooth compactly supported tight wavelet frames with three generators in . Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.

COMPACTLY SUPPORTED MULTIVARIATE, PAIRS OF DUAL WAVELET FRAMES OBTAINED BY CONVOLUTION

2008 ◽  
Vol 06 (02) ◽  
pp. 183-208 ◽  
Author(s):  
MARTIN EHLER
Keyword(s):  
Approximation Order ◽  
Refinable Functions ◽  
Wavelet Frames ◽  
Refinable Function ◽  
Vanishing Moments ◽  
Wavelet System ◽  
Compactly Supported ◽  
Mask Size ◽  
Dual Wavelet Frames ◽  
Dual Wavelet

In this paper, we present a construction of compactly supported multivariate pairs of dual wavelet frames. The approach is based on the convolution of two refinable distributions. We obtain smooth wavelets with any preassigned number of vanishing moments. Their underlying refinable function is fundamental. In the examples, we obtain symmetric wavelets with small support from optimal refinable functions, i.e. the refinable function has minimal mask size with respect to smoothness and approximation order of its generated multiresolution analysis. The wavelet system has maximal approximation order with respect to the underlying refinable function.

scholarly journals On Parseval Wavelet Frames via Multiresolution Analyses in

2019 ◽  
Vol 63 (1) ◽  
pp. 157-172
Author(s):  
A. San Antolín
Keyword(s):  
Fourier Transform ◽  
Multiresolution Analysis ◽  
Extension Principle ◽  
Refinable Functions ◽  
Wavelet Frames ◽  
Approximate Continuity ◽  
Reducing Subspaces ◽  
The Fourier Transform ◽  
Frame Multiresolution Analysis

AbstractWe give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

scholarly journals Simultaneous Contour Method of a Trigonometric Integral by Prudnikov et al.

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1453
Author(s):  
Robert Reynolds ◽  
Allan Stauffer
Keyword(s):  
Closed Form ◽  
Contour Method ◽  
Substantial Part ◽  
Exponential Functions ◽  
Closed Form Solutions ◽  
Definite Integrals ◽  
Transcendental Functions ◽  
Trigonometric Integral

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

2014 ◽  
Vol 57 (2) ◽  
pp. 254-263 ◽  
Author(s):  
Ole Christensen ◽  
Hong Oh Kim ◽  
Rae Young Kim
Keyword(s):  
Extension Principle ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Number Of Generators ◽  
Unitary Extension

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

scholarly journals Unitary Extension Principle for Nonuniform Wavelet Frames in L2(ℝ)

2021 ◽  
Vol 17 (1) ◽  
pp. 79-94
Author(s):  
Hari Krishan Malhotra ◽  
◽  
Lalit Kumar Vashisht ◽  
Keyword(s):  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension

scholarly journals Minimum-Energy Bivariate Wavelet Frame with Arbitrary Dilation Matrix

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fengjuan Zhu ◽  
Qiufu Li ◽  
Yongdong Huang
Keyword(s):  
Scaling Function ◽  
Minimum Energy ◽  
Sufficient Conditions ◽  
Reconstruction Algorithm ◽  
Wavelet Function ◽  
Necessary Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Numerical Examples ◽  
Dilation Matrix

In order to characterize the bivariate signals, minimum-energy bivariate wavelet frames with arbitrary dilation matrix are studied, which are based on superiority of the minimum-energy frame and the significant properties of bivariate wavelet. Firstly, the concept of minimum-energy bivariate wavelet frame is defined, and its equivalent characterizations and a necessary condition are presented. Secondly, based on polyphase form of symbol functions of scaling function and wavelet function, two sufficient conditions and an explicit constructed method are given. Finally, the decomposition algorithm, reconstruction algorithm, and numerical examples are designed.

Modal Analysis of Multi-Stepped Distributed-Parameter Rotor-Bearing Systems Using Exact Dynamic Elements

2001 ◽  
Vol 123 (3) ◽  
pp. 401-403 ◽  
Author(s):  
Seong-Wook Hong ◽  
Jong-Heuck Park
Keyword(s):  
Modal Analysis ◽  
Closed Form ◽  
Complete Analysis ◽  
Distributed Parameter ◽  
Analysis Method ◽  
Closed Form Solutions ◽  
Analysis Scheme ◽  
Rotor Bearing ◽  
The Matrix ◽  
Transcendental Functions

Although the exact dynamic elements have been suggested by the authors [1] and proved to be useful for the dynamic analysis of distributed-parameter rotor-bearing systems, difficulty remains in computation because of the presence of transcendental functions in the matrix. This paper proposes a complete analysis scheme for the exact dynamic elements, a generalized modal analysis method, to obtain exact and closed form solutions of time and frequency domain responses for multi-stepped distributed-parameter rotor-bearing systems. A numerical example is provided for validating the proposed method.

Sign in / Sign up

Export Citation Format

Share Document