Multivariate Compactly Supported Fundamental Refinable Functions, Duals, and Biorthogonal Wavelets

1999 ◽  
Vol 102 (2) ◽  
pp. 173-204 ◽  
Author(s):  
Hui Ji ◽  
Sherman D. Riemenschneider ◽  
Zuowei Shen
Keyword(s):  
Refinable Functions ◽  
Biorthogonal Wavelets ◽  
Compactly Supported

scholarly journals A SINISTER VIEW OF DILATION EQUATIONS

2005 ◽  
Vol 03 (01) ◽  
pp. 67-77
Author(s):  
DAVID MALONE
Keyword(s):  
Refinable Functions ◽  
Refinable Function ◽  
Compactly Supported ◽  
Rate Of Growth ◽  
Dilation Equation ◽  
Dilation Equations

We present a technique for studying refinable functions which are compactly supported. Refinable functions satisfy dilation equations and this technique focuses on the implications of the dilation equation at the edges of the support of the refinable function. This method is fruitful, producing new results regarding existence, uniqueness, smoothness and rate of growth of refinable functions.

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 Construction of Compactly Supported Refinable Componentwise Polynomial Functions inℝ2

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Yanmei Xue ◽  
Ning Bi
Keyword(s):  
Iteration Algorithm ◽  
Sufficient Condition ◽  
Polynomial Functions ◽  
Refinable Functions ◽  
Compactly Supported ◽  
Polynomial Property

We provide a sufficient condition for constructing a class of compactly supported refinable functions with componentwise polynomial property inℝ2. An iteration algorithm is developed to compute the polynomial on each component of the functions' support. Finally, two examples for constructing the symmetric refinable componentwise polynomial functions are given.

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.

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